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2Tracing the Modern Concept of Convexity*
Sjur D. Flám1** and Gabriele H. Greco2
1 Department of Economics,
University of Bergen 5007 Norway E-mail: sjur.flaam@econ.uib.no
2 Dipartimento di Mathematica, Universita delgi Studi di Trento,
I-38050 Povo (Trento) Italia;
E-mail: greco@science.unitn.it.
Abstract Several manifestations of convexity were studied already in the antiquity. The modern concept emerged, however, only around 1900, notably in the works of Peano and Minkowski. This motivates us to review here some main contributions of the latter. In doing so, we attempt to offer a friendly invitation to the history and concepts of convex analysis. Our emphasis is on convex bodies, extreme points, separation, Minkowski functionals, supports and polarity.
1. Background and Motivation
The notion of convexity dates back to Archimedes at least, but receded, during the birth and early growth of calculus, into the background. The last half century has seen a surge of interest though. Convexity, and the analysis of its many manifestations, now holds much prestige and offers wide applicability; see e.g. (Borwein and Lewis, 2000), (Hiriart-Urruty and Lemarechal, 1993), (Hormander, 1994), (Rockafellar, 1970), (Tikhomirov, 1980).
The rapid development of theory motivates us to reconsider briefly the historical introduction of some key concepts. In doing so, we limit attention here to some chief contributions made by the pioneer Hermann Minkowski (1864-1909), referred to as M for short. We shall review none of M’s many results outside convex analysis - be it in number theory or physics1. And even inside that field we select just a few, namely those that concern the interplay between geometry and optimization, excluding here discrete structures. Of great interest to such interplay -and to extensions of convex analysis (Aubin and Frankowska, 1990), (Clarke, 1983), (Clarke et al.), (Rockafellar and Wets, 1998) - are several interwoven facts and features. Most of them relate to:
• the algebraic and topological properties of convex sets - and especially of polyhedral instances;
• the characterization of such sets by functional counterparts such as gauges or supports; and finally,
• ”From Germany, after an enquiry through the Commissariat of Enlightenment, Persikov was sent three parcels containing mirrors, convexo-convex, concavo-concave and even some sort of convexo-concave ground glasses”(Bulgakov, 2003).
** Thanks for generous support are due Ruhrgas, Meltzers h0yskolefond, R0wdes Stiftelse, Universita delgi Studi di Trento, and CNR, Italy.
1 The exclusion of number theoretic results is warranted, we feel, given the treatise (Hancock,1939).
• the dual description and polarity of these objects that stems from separation theorems.
These various faces of convexity, already identified by M, are the ones we shall emphasize. Thus, M’s pioneering studies of volumes, areas, curvatures, and closest packings of convex sets will get no mention. In particular, we keep silent on the Brunn-Minkowski theory (and related isoperimetric or differential-geometric properties) of convex bodies.
It seems fair to claim that convex analysis stems mainly from mechanics and geometry; see (Fenchel, 1983).2 However, since M, certainly a founding father, took little part in these fields, the said claim is not quite correct.3 In fact, he came indirectly, repeatedly - and rather surprisingly - to convex analysis from number theory and by a geometric approach to the latter.4 In reviewing some features of that approach we try not to regard results with hindsight. Rather we attempt to appreciate key theorems - and sometimes their proofs - within the mathematical culture that reigned, say around 1895. At places it appears natural though, to mention some later developments briefly. Also, we record how the notion of convexity evolved in the hands of M, a particular feature being that he started out from a most important class of convex functions - and not from corresponding sets.
Throughout we seek to stress the originality of selected results - and to make clear how modern M was in his approach. We address several sorts of readers. Expertise in convex analysis is not needed. In fact, the subsequent material should interest diverse people be they students or scholars of analysis, optimization, geometry, economics, or the history of mathematics.
Our reading of Minkowski’s works were focused mainly on Geometrie der Zahlen (1896) - GZ for short - and the two chief chapters XXII and XXV in Gesammelte Abhandlungen (1911) - henceforth called GA.5 Unless stated otherwise each subsequent reference to pages concern GA vol. II. All theorems singled out with bold letters in the main text below are attributed to M.
For a start, we found it inviting to read M’s abstract of GZ in a letter to Hermite (GA, chap. XI):
2 In their excellent book Borwein and Lewis (2000) say that ” convex analysis developed historically from the calculus of variations..” This assertion seems justifiable only up to the extent that isoperimetric problems reside at the heart of the two fields. Giaquinta and Hildebrandt (Giaquinta and Hildebrandt, 1996) opinion though, that: ”Amazingly, convex functionals do not play any role in the classical calculus of variations before the turn of the century, although they seemed predestinated, to assume a central place. One has to realize that only Minkowski recognized the notions of convex set and convex function to be central concepts in mathematics, although the concepts of a convex curve and a convex surface were well-known and often used in ancient times; then convexity meant ’’locally eggshaped.”
3 It may mirror a natural tendency, in any field, to appropriate non-intentional contributors provided they be great.
4 For example, concerning approximation of numbers by rationals, M says in the preface to (Minkowski, (1896) ): ” Ich bin zu meinen Satzen durch räumliche Anschauung gekommen.” In a letter December 1887 to Hilbert he says: ”Ich bin auch ganz Geometer geworden, und bedaurere aus diesem Grunde doppelt, nicht in Ihrem Kreise weilen zu kännen.” (Minkowski, 1970).
5 GA were published posthumously. Omitted there is GZ and another number theoretic book (Minkowski, 1927) - as well as studies of the relativity principle, written by M alone or with Lorenz and Einstein.
2. Minkowki’s Partial Resume
”The largest part of the book (GZ) deals with functions in n variables x1, x2,xn, which, like the square root of a positive quadratic form, satisfy the conditions
{<^>(xi, X2, ...,xn) > 0, unless xi = 0,x2 = 0, ...,xn = 0,
^(0, 0,..., 0) = 0,
^(ixi, tx2,..., txn) = t<^(xi, x2,..., xn) si t > 0,
(B) <^(xi + yi,x2 + V2 ,...,xn + Vn) < <^(xi ,x2 ,...,xn) + <^(vi ,V2 ,...,Vn),
(C) ^(-xi, -x2,..., -xn) = <^(xi ,x2,...,xn).
Let £i ,£2 ,...,CV be linear forms with real coefficients in variables xi ,x2 ,...,xn, and among these let n forms have determinant different from zero. Denote by ^(xi, x2,..., xn) the maximum of the absolute values of £i, £2,..., £v. Evidently, such a function <P satisfies the conditions of a function <^>. I establish the theorem:
When solves (A), (B), (C), and 6 is any positive quantity, one may always find a function ^, as just characterized, such that, for all possible xi ,x2, ...,xn one has
1 < ^/^ < 1 + 6. (1)
As a result, the intgral / J .../ dxi dx2 ...dxn over the domain ^(xi ,x2 ,...,xn) < 1 has always a definite value. Denote by J that value. I show that one can always find integers xi,x2 ,...,xn such that 0 < <^(xi ,x2,...,xn) < 2/Ji/n.”
Prominent already here are positive sublinear functionals, finitely generated convex sets, polyhedral approximation, convex bodies, and integer solutions. It all started, however, with
3. Quadratic Forms and Integer Lattices
Positive quadratic forms were central in M’s first number theoretic studies. Brief mention of two early works suffices to bring this out. One work - his very first: GA, chapt. I, which he wrote at the age of 17 - more than solved the award-winning problem, stated by the French Academy of Sciences, to represent any natural number as the sum of five squares. As vehicle M used quadratic forms with integer coefficients - and undertook to classify these. The geometric properties of positive quadratic forms, and notably their convex lower level-sets (so-called ellipsoids), then occupied center stage as smooth instances of convex bodies. The other early work: GA, chapt. XIX - on closest packing of such bodies - brought M to study arithmetic equivalence and reduced versions of positive quadratic forms (see also GA, chapt. XXI). In that enterprise he showed that the latter objects constitute a convex cone generated by finitely many extreme rays.
For our review we must first fix some notations. Although M frequently took the three-dimensional space as the ambient setting, equipped with its customary Grassmannian inner product, the results considered below extend verbatim to finitedimensional spaces - and often beyond (Holmes, 1975).6 For this reason, and to facilitate reading, we prefer the modern (coordinate-free) setting of a Euclidean space E, endowed with inner product (■, ■) and associated norm ||-|| . Regarding that space our symbols are standard: Let a//C, bdC, dC, convC, dimC, intC,
6 Kothe (Kothe, 1969) says : ’’Provided that necessary care is taken, methods that go back to Minkowski can also be applied to convex sets in vector spaces of infinite dimension.”
denote the affine hull, boundary, closure, convex envelope, dimension, interior and (d-dimensional) volume respectively of C C E. When C, D C E and R C R, we write C + RD for the Minkowski combination {c + rd : c G C, d G D,r G R}. In particular, given two points x,y G E we shall deal with closed segments [x,y] := {x} + [0,1] {y — x} and half-open versions [x, y) := {x} + [0,1) {y — x}. I always denotes a nonempty finite set of indices, and R1 is the set of all functions x : I ^ R.
As said, positive quadratic forms were main objects in M’s first papers. The mapping x ^ (x, Qx)i/2 is then positively homogeneous, subadditive, non-negative, and vanishes only at the origin. M calls functionals y : R1 ^ R that satisfy precisely these properties, einhellige Strahldistanzen. This special class of convex functions, now named Minkowski functionals, constitute his point of departure in GZ. He shows there that each of them is continuous, bounded below (and above) by a suitably scaled version of maxiGl |xj| , the relevant factors being called Distanzcoefficienten; see also (3).
Much of M’s motivation for considering such functionals stems from Hermite’s preceding concern with the smallest number r for which (x, Qx)i/2 = r is solvable in integers, that is, the minimal level r such that the ellipsoid |x : (x, Qx)i/2 < r j
intersects the integer lattice Z1. This problem led M to translate a symmetric convex ground set C = {x : y(x) < r} by every point in Z1. More generally, any basis B in (the finite-dimensional Euclidean) space E generates a complete lattice L := beB Zb whose fundamental mesh, the parallelepiped ^beB [0,1] b, has vol-
ume u Then it holds
Theorem (on lattice points)7. Let C be symmetric convex and L a complete lattice with mesh-volume u Suppose volC > 2dlmeu. Then C contains at least one nonzero lattice point. □
Corollary. For each j G I let a.jx := ieI a^-xj be a linear form and Cj a positive number such that ffjCj > |det(ajj)| > 0. Then there exists z = (zj) G Z1 such that |a.jz | < Cj for all j. □
Only some properties of ellipsoids were crucial here, namely those that led him to identify the notion of convex surfaces and bodies as fundamental:
4. Convex Surfaces, Bodies and Sets
Given a positive, sublinear 7, M singles out the corresponding surface (Aichflache) := {x : y(x) = 1} together with the associated body (Aichkorper):
Cy := {x : Y(x) < 1} , (2)
and shows that the latter is stable with respect to formation of segments; that is, x,y G CY ^ [x,y] C CY. He announces in GZ §8 - and proves in §16 - that is supported in each point by a hyperplane.
Thus, M starts from positive sublinear functions Y and proceeds to consider their representative level sets and bodies CY. Only thereafter does he follow a
7 This theorem resides at the foundation of essential parts of algebraic number theory (Ribenboim, 2001).
long geometric tradition - one that dates back to Archimedes at least - according to which convexity is a property displayed by certain curves and surfaces. Intuitively, a convex surface should never dent inward; that is, any chord should reside on its ”inner” side. Equivalently, any ’’tangent” should be ’’external.” So, quite naturally, when M introduces a convex surface - encompassing an ’’interior point”, say 0 - as primitive object, he states the
Definition (convex surfaces GZ §17). A set S C E is declared a nowhere concave surface around 0 iff first, each ray from 0 contains at least one point in S \ 0; and second, through each point in S there must pass at least one supporting hyperplane (Stutzebene), this meaning that S lies fully on one side (in one halfspace). □
M argues that 0 G S; otherwise any line, not contained in a supporting hyperplane through 0, would intersect S on both sides of that plane - a contradiction. Similarly, each ray from 0 has exactly one point in common with S.
Letting ys (x) := inf {r > 0 : x G rS} M establishes a one-to-one correspondence between surfaces of the said sort and positive sublinear functionals via S = {x G E : ys (x) = 1} .8
Such a surface S generates a convex body C := [0,1] S with bdC = S.9 Thus emerges also a bijection C ^ S between convex bodies and their surfaces. This approach, making convex bodies enter second - either as a lower level {x : ys (x) < 1} or as an envelope conv {0 US} - was turned around in M’s subsequent, more modern definition. Convexity is tested there not by means of barycenters - as did Archimedes, or by supporting hyperplanes as did M first - but simply by onedimensional sections:
Definition. A set C C E is called a convex body when primo, every line has a closed interval, a point, or nothing in common with C, and secondo, C does not belong to a proper subspace.i0 □
In passing, it deserves notice that sets, and especially Cantor’s set theory, around 1900 still received less than full respect within the mathematical community. Notably Kronecker, one of M’s prominent teachers, conveyed lack of enthusiasm for Cantor’s work. Hilbert (Hilbert and Minkowski, 1911) exaggerates somewhat though, in claiming that M ”was the first mathematician of our generation..who recognized the great significance of Cantor’s theory..”ii Also worth mention is that the basic notions of topology and linear algebra were recently conceived, or in their early infancy, at M’s time and neither commonly known nor widely used.
8 GZ p. 36. Of particular interest is the p-norm j(x) = (^2ieI \xi\p) see GZ §18.
9 The notion nowhere concave surface, used in GZ, is replaced by convex surface in GA vol.II. p. 123.
10 See GA p. 131 and 103. M proves that the barycenter fC xdx/volC of a convex body belongs to its interior (GA 144); on this see also (Bonnesen and Fenchel, 1934).
11 Hilbert’s assertion seems appropriate for the two of them. But it hardly fits even his own professional environment. Indeed, A. Schonflies, a collegue of Hilbert in Gottingen, wrote on set theory in (Encyklopadie der Mathematischen Wissenschaften), vol.I, part 1, Ch.5. And Cantor’s influence had long been great in France and Italy. M and Hilbert were collegues from 1902-1908 in Gottingen; see (Reid, 1962). Hilbert says in his Gedachtnisrede: ”Er war mir ein Geschenk des Himmels...” (GA, vol. I, p.XXXI).
Regarding the definition above, M immediately brings out that convexity - and thereby also closure and boundedness - derives from the corresponding property being valid along any line. His argument goes as follows: Since C has full dimension, it contains a simplex whence an interior point (GA p. 131).i2 Fix any point of that sort as the origin; employ orthogonal coordinates to have E = R1 for some finite set I; and define an associated gauge function this way: Along an arbitrary ray, emanating from the chosen origin, let 0, x0 denote the largest closed line segment contained in C. For any other point x = (x*) on that same ray, each ratio xj/x0,* G I, must equal a common number yC(x) > 0.is
Clearly, the function yC so defined satisfies C = {x : yC (x) < 1}. For an alternative definition of yC , also due to M and now standard, see (12). Upon verifying that yC is positively homogeneous, subadditive - and thus continuous - M concludes that C must be closed. Letting M and m > 0 denote the attained maximum and minimum respectively of yC on the unit sphere, the positive homogeneity yields
m ||x|| < yC(x) < M ||x|| for all x. (3)
Consequently, C, being contained in a ball of radius M, must be bounded as well. Clearly, in current jargon, yC defines a norm - and an associated geometry where the standard body {x : yC (x) < 1} takes the place of the customary closed unit ball. M also mentions almost explicitly (GA p. 136) that x G intC, y G C ^ [x, y) C intC, a property now named linear accessibility.i4 His arguments in GA p. 161-2 bring out that C = c1(intC), and conversely, if C equals the closure of an open convex set O, then O = intC.
The priority M assigns to convex bodies reflects, of course, the convenience of situating his geometric objects within a minimal topological context, one which makes for coincidence between relative and absolute interiors.i5 That is, he prefers to arrange situations so that the algebraic interior
coreC := {c G C : R+(C — c) = E} (4)
equals the topological interior. In finite dimensions, when C is convex, intC = coreC. More generally, intC C coreC, and what imports in manifold settings is
12 In modern notation, M brings out here that for any simplex in finite dimesions, and more generally for any convex body C, relintC = coreC = intC.
13 See GA p. 132. More generally, for any two vectors x = (xi) ,y = (yi) on the same ray must satisfy xi/yi = yc(x)/yc(y) provided yi = 0. Then, if yc(x) = 1 and ||y|| = 1, 1/yc (y) is the distance of x to the origin; see GA p. 134.
14 As said, the avenue just described runs opposite to M’s first approach to convex sets in GZ §1-8. There he introduced positively homogeneous "radial distances” (Strahldistanzen) S(x — x') > 0 along rays. These should satisfy S(x — x') =0 x = x', but not necessarily the triangle inequality. A function of this sort is fully determined by any of its lower level sets {x : S(x — x') < r}, the reference point x' being arbitrary and r > 0. When S(-) is continuous, M calls such sets starshaped bodies (Strahlenkorper). A vector collection C, having nonempty interior, is a convex body iff starshaped with respected to any interior point. Thus, starshaped bodies are convex iff S(-) is a norm (ein einhelliger Strahldistanz; see (Minkowski, (1896) , § 24)). In addition, S is symmetric (wechselseitig) iff C is so; that is, S(x) = S(—x) iff x £ C ^ —x G C.
15 When intC = 0, one may invoke relative interior as described in (Rockafellar, 1970) and (Rockafellar and Wets, 1998).
to have at least one of these sets nonempty. Anyhow, M certainly accommodates convex sets which have empty interior or are unbounded, saying that those he mostly considers have three defining properties: First, x,y G C ^ [x, y] C C; second, C is bounded; and third, it must be closed (GA p. 106 and 134 ). Examples of such objects, says M, include (the convex polytope (Grunbaum, 1967))
P :=J ^ Ajxj : ^ Aj = 1, VAj > 0! , (5)
Uel jGl )
with I finite. The set P so defined must be part of any convex set that comprises xj (GA p. 135). The last observation amounts, of course, to say that the convex hull of a set, formed, by convex combinations of its members, equals the intersection of all convex supersets. In particular, P = conv {xj : i G /} . Thus M provides the customary direct, primal, algebraic-geometric description of a convex set in terms of its habitat. For a minimal description of polytopes M notes that if xj = ^jGl a*xj G P with Aj < 1, then P = conv {xj : i G I\j} ; see GA p. 135-136.i6. Repeated elimination leaves P finally as the convex hull of
5. Extreme Points
Regarding the set P in (5) M declares xj a corner point (Eckpunkt) if it does not belong to the convex hull of xj, j = i, and he adds:
Definition. ”A line through a corner point cannot contain points from P on both sides”(GA p. 136). □
Corner points are precisely those he later calls extreme, that is, those elements x in a convex set C for which C\x remains convex. As a prelude to his characterization of compact convex sets, M notes that P equals the convex hull of its corner points (GA p. 135-136).i7 M devotes an entire section to this general notion (GA §12, p. 157-161). Established there is the ’’reconstruction” of a compact convex set from its subset extC of extreme points:
Theorem (extreme points GA p. 160). Any compact convex subset of a finitedimensional vector space equals the convex hull of its extreme points: C = conv(extC). i8 □
In essence, M’s proof of this important result has become the standard, inductive one which invokes that any point extreme in the intersection with a supporting hyperplane must also be extreme in the original set. (Clearly, the assumption that intC be non-empty, is superfluous.)
M demonstrates that any convex body C, containing 0 in its interior, can be closely approximated, from inside and outside, by two homothetic polytopes (GA
16 In other words: A compact convex set P is a polytope if extP is finite.
17 If P, as defined above (5), contains interior points, it is named a polyeder. That notion differs from the modern polyhedron, meaning an intersection of finitely many closed halfspaces. Put differently: M’s polyeder is a bounded polyhedron with nonempty interior.
18 Caratheodory’s sharper form says that at most dim E + 1 extreme points are needed.
p. 138-139). More precisely, given any 8 > 0 there exists a set P of type (5) such that
P C C C (1 + 8)P. (6)
This result, used for his study of volumes (GA p. 124), he later refines, showing that P can be generated exclusively by extreme points of C (GA p. 160-161). Approximation (6) relates closely to his number theoretic studies. To wit, (1) is a functional counterpart to set-theoretic version (6) - and also similar to (3). (Corresponding smooth approximations P and <P are displayed in GA p. 233-235.)
M’s extreme point theorem has become a fundamental tool of functional analysis. There it is referred to as a the Krein-Milman theorem (Krein and Milman, 1940), saying that a compact convex subset of a locally convex separated space equals the closed convex hull of its extreme points. (For a proof and historical references see (Valentine, 1964).) Also intimately related is Choquet’s theorem: Any element of a metrizable compact convex set in a locally convex separated space is the barycenter of a Borel probability measure supported, by the extreme points of the said set; see Phelps (Phelps, 2001).
Returning to finite dimensions, and accommodating the set rextC of extreme rays, the compactness assumption may be dropped to have Klee’s theorem (Klee, 1958): For a line-free closed convex set C it holds that C = conv(extC U rextC). Of particular importance are instances where both sets extC and rextC are finite:
6. Finitely Generated Sets
The finitely generated, bounded set P, defined in (5), is nowadays called a convex polytope. The combinatorial properties of these objects have long attracted much interest. Fundamental in that regard is the
Theorem (on polytopes, GA, chapt. XXII; see (Grunbaum, 1967)). (i) Let P be a, d-dimensional polytope. Then, each maximal proper face (ie, facet) F of P has a unique outward normal vector vf with ||vF|| = vol^_iF, such that
F = F' ^ vF G R+vF'; dim a// {vF} = d, and vF = 0. (7)
(ii) Conversely, given a finite system of vectors {vF} satisfying (7), there exists a d-dimensional polytope P - unique up to translation - having outward normals vf to facets F with ||vF|| = vold_iF.
We find M’s proof of (ii) particularly interesting. An outline goes as follows: Clearly, for appropriate r = (rF) > 0, the polytope P(r) := HF j*: ^ y, x^ <rp j is the natural candidate. Since, by the Brunn-Minkowski theorem,i9 0 < r ^ ^(r) := [volP(r)]i/d is concave, continuous and homogenous, the (hypographical) set {(r,w) > 0 : ^(r) > w} must be a closed convex cone. That cone has a compact convex ”base” B = {r > 0 : ^(r) = 1} over which the linear form r F l|vFII rF
19 The Brunn-Minkowski theorem: If C0,C1 C E are closed convex and Cr : = (1 — r)C0 + rC1; then the function [0, 1] 3 r ^ (volCr)1/ dlmE is concave, and strictly so unless C0, C1 are homothetic or lie in parallel hyperplanes. For proofs see (Bonnesen and Fenchel, 1934).
attains its minimum at some r*. Then P(r*) has a vector of facet volumes proportional to (||vF ||) so, after suitable scaling the desired conclusion obtains.
One ’’evident” but important feature comes up in connection with polytopes, namely: The topological closure derives from purely algebraic assumptions. In that regard M, while dealing almost exclusively with closed sets, notes that if P has non-empty interior, then each convex combination in (5), having strictly positive coefficients, must be interior (ein innerer Punkt). Otherwise, when miP = 0, such a combination is inwendig, i.e. it belongs to the relative interior (GA p. 136).
As said, M mostly dealt with bounded closed convex sets, (5) being one example. Important unbounded instances include linear inequality systems Ax < 0, featuring a matrix A of size I x J, the solution set
K := {x€ RJ : Ax € R- } (8)
of which must be not only closed convex but also a cone, that is to say, rK C K for all real r > 0; see GZ §19.20 The geometric theory of linear inequalities thus begins with M (and was later developed systematically by Motzkin (Motzkin, 1936)). M finds that K is formed by all non-negative linear combinations of finitely many so-called extreme directions (ausserste Losung):
Theorem (on polyhedral cones GZ §19): A polyhedral cone, as defined in (8) is generated by finitely many rays. □
Conversely, Weyl (Weyl, 1935), in adding to this ’elementary” characterization, showed that a finitely generated cone must be polyhedral.21 As above, a topological proposition thus derives from purely algebraic assumptions, namely: Any finitely generated cone must be closed.
Noteworthy is the key role assigned by M and Weyl to extreme supports (extreme Stütze/ Stützebenen).22 M defines these objects as supporting hyperplanes that correspond (via polarity) to extreme points of the polar set (see GA p. 164 and Section 8 below). So, in the present context, the normal vectors to extremal supports of the polar cone
K- := K0 := {x* : (x* ,x) < 0, Vx € K} (9)
constitute the extreme directions that generate K. Anyway, if the linear homogeneous inequality system Ax < 0 admits non-zero solutions, then, geometrically speaking, these constitute an unbounded pyramid with apex at 0. This result dictates minimal descriptions of polyhedral cones - either internal-primal in terms of extreme directions or external-dual by means of extreme supports.
These results on cones point to Farkas’ Lemma (Farkas, 1902): If for some c € RJ, the system —x € K, (c, —x) > 0 is inconsistent; that is, if the inequalities
Ax > 0, (c, x) < 0 (10)
20 M made inhomogeneous systems Ax < b,b = 0, homogenous by adjoining a new variable - a technique which later has become standard.
21 For a new proof of both statements see (Rockafellar and Wets, 1998) 3.52.
22 In §14 p.166-168 M characterizes an extreme support as what he names a Tan-
gentialeben (a tangent plane). The latter object is a supporting plane HE : = {x : (x*,x) = S**(x*) — e} , with e = 0, such that the parallel intersection C n HE, for e > 0 sufficiently small, contains a ball (of dimension one less than C) with radius p(e) such that p(e)/e ^ as e ^ 0.
have no solution, then c must belong to the polar cone K0 generated by the rows aj. of A (see (9) and Section 8). This means that the system
admits a solution.
Farkas’ Lemma helps to emphasize the key role of convexity in combinatorial optimization (Bachem,1983 ), (Schrijver, 2003). It also serves as the point of departure in modern theory of finance (Duffie, 1992). Specifically, suppose security j G J is available right now, in any quantity xj G R, at unit price cj, and promises payoff ajj tomorrow if state i G I then comes about. A portfolio or investment strategy x G RJ in securities which solves (10), is called an arbitrage (Ellerman, 1984). That investment requires negative purchase cost (by offering an immediate bonus |(c, x) | > 0), and it yields at least 0 payoff whatever happens tomorrow. Such opportunities cannot reasonably persist in an equilibrated market. Thus, an equilibrium price system c must solve (11).
The Minkowski-Weyl algebraic-geometric characterization of polyhedral cones fits to a large, now classic field concerned with the solvability and geometry of linear/affine inequality/equality systems (Motzkin, 1936). Incorporated there is a long list of theorems on alternatives (De Giuli et al. ), including that of Gordan (Gordan, 1873), frequently used to prove Fritz-John type optimality conditions for constrained programs (Borwein and Lewis, 2000). Linear programming aroused interest in those theorems because they bear on the feasible set. Conversely, they are often proved via linear programming duality (Chvatal, 1983).
A subset of an Euclidean space is nowadays called finitely generated if it equals the Minkowski sum P + K of a polytope P and a finitely generated cone K. Concerning these objects there is a theorem attributed to Minkowski-Weyl: A set is finitely generated iff polyhedral.
So, a minimal description of finitely generated sets comes either via extreme points/directions or supports; see (Bazaraa and Shetty, 1979). Defining a function / : E ^ RU {+^} to be polyhedral (or finitely generated) iff its epigraph epi/ := {(x,r) : /(x) < r} is of that special sort, we arrive at most important objects of mathematical programming: namely, functions / whose effective domains {x : /(x) < +^} are polyhedral and, on that set, equal the pointwise maximum of finitely many affine functions.
Convex sets that cannot be finitely generated are often best described by
7. Gauges, Supports and Tangent Cones
When 0 is interior to the convex body C, M shows that the associated gauge - also called the Minkowski (gauge) functional -
x ^ Yc (x) := min {r > 0 : x G rC} ...( = min {r > 0 : x G rbdC}) (12)
vanishes only at the origin, is well defined, finite-valued, positively homogenous, and subadditive whence continuous.23
23 When E = R1 with finite I, and x = (xi), the subadditivity entails yC(x) <
X^gi YC(xiei), see GA p. 132-133. Letting r := maxigi yc(±ei), it follows that
(11)
Yc(x) < r£t6l \xí\ and thereby ja; : Ha;^ <
C {x : ||a;|| 1 < ^} C C.
Function (12) has a constructive, clear-cut, geometric meaning, namely: yC (x) tells exactly how much must C be inflated/deflated in order to just contain x. It was not common practice at M’s time though, to define or use functions, like (12), that assume no explicit, closed form. The originality of M’s approach is underlined by his alternative, more axiomatic definition of gauge (12): yc is the positively homogeneous function that equals 1 on the boundary of a convex body C with 0 € intC.
When C is symmetric, (12) defines a norm x ^ yc (x) on E, with associated metric (distance) d(x,y) := yC (x — y) and closed unit ball C. A vista then opens up for other geometries than that of Euclid. Finite-dimensional vector spaces normed in this manner are often called Minkowski spaces (Valentine, 1964). More generally, construction (12) makes it natural to declare a topological vector space normable if it contains a convex body (that is, a bounded closed convex set with nonempty interior).
Conversely, given a function x ^ y(x) satisfying the properties just mentioned, its associated lower level set (2) must be a convex body, containing 0 in its interior, and satisfy yC = Y :
Theorem (on gauges). Via (12) and (2) there is a one-to-one correspondence C ^ yc between two special classes of quite different nature: on one side compact convex sets C C E which contain 0 as interior point; on the other side non-negative, positively homogenous, subadditive, continuous functions y = Yc : E ^ R that vanish only at the origin. □
Within the said classes a monotonicity property evidently holds: C C C' yc >
YC'. Also, YrC = Yc/r whenever r > 0, and for any finite family of convex bodies Cj, i € I, each containing 0 as interior point, we have YconvUic = maxj yc; see GA p. 157.
The gauge of a convex body C with 0 € intC, provides a unit of measurement along every 1-dimensional linear subspace. Orthogonal to such a subspace stands a so-called hyperplane. Therefore, instead of asking how C fares on lines, one may inquire how it relates to hyperplanes. Pursuing this dual perspective M introduces the support function (Stützebenenfunktion) S* of C; see GA p. 4-6 and p. 144-147. The requirement that 0 € iniC is later dropped, GA p. 150-153. Quite generally, he posits
S*(x*) := sup (x*,x) = sup{(x*,x) — SC(x) : x € E} (13)
albeit without introducing the more modern, extended indicator x ^ SC (x) which equals 0 on C and elsewhere. He observes that S* (■) so defined becomes positively homogeneous, subadditive. In particular, 0 € C S* > 0, and S* vanishes only at the origin iff 0 € intC. Also, S*(x*)+ S*(—x*) > 0 for all nonzero x* iff intC is nonempty. Otherwise, if S* (x*) +S* (—x*) = 0 for some nonzero x*, then C is part of the hyperplane {x : (x*,x) = ¿¿(x*)}. A symmetric (balanced) set K := c~c, with 0 as center (Mittelpunkt GA p. 4), has symmetric support s* (■) = sk (-■).
More generally, its suffices for the sublinearity of yc that C be convex and 0 G coreC. Then coreC = {x : YC (x) < 1} C C C {x : YC (x) < 1} . If moreover, C is symmetric, then YC becomes a seminorm. These are instrumental for the definition of topologies in linear spaces (Holmes, 1975). Note that YC(x) = 0 iff R+x C C.
Most importantly, M records that the original set can be recovered, namely
C = {x : (x*,x) < 8C(x*), Vx*} , (14)
see GA p. 145. Thus, once again M identifies two sides of the same convex coin:
Theorem (on support functions). Via (13) and (14) there is a one-one correspondence C ^ 8C, this time between larger classes: on one side closed convex sets, on the other side positively homogeneous, subadditive functions. □
(14) provides a dual description of a convex set by means of objects from the conjugate space E* (= E), consisting of all continuous linear functionals x ^ x*(x) = (x*,x) G R. Clearly, monotonicity again prevails: C C C' ^ 8C < 8**/. Also, 8*C = r8C whenever r > 0, and for any finite family of compact sets C*, i G I, one has 8*onvUiC. = maxj 8**.; see GA p. 156.
The possibility to codify a closed convex set by its support function is often very useful. For an economic example, let a price-taking (also called perfectly competitive) firm be fully described by a closed nonempty convex set C, representing its technology and comprising all feasible input-output vectors. That firm can equally well - and uniquely - be depicted by its profit function x* ^ 8C(x*), reporting the maximal profit obtainable under diverse price regimes x* G E*; see (Diewert, 1981). Clearly, given a finite set of observations [x*j, 8C (x*j)] , i G I, then econometrically speaking, C is ’’overestimated” by the possibly larger set nj£/ {x : (x*j ,x) < 8C (x*j)} .
Other examples come from nonsmooth analysis, dealing with generalized derivatives and convex-valued correspondences. Specifically, consider the two directional derivatives
M/ J\ l- /(x + ^x + td) — /(x + ^x) /i r\
/ (x; d) := lim sup -------------------------------------- (15)
^x^0,t^0+ t
and
c ( f(x + tAx + td) - f(x + tAx) \
/ (x; a) := sup I lim sup ------------------------------------- , (16)
V t^0+ t J
introduced by Clarke (Clarke, 1983) and Michel-Penot (Michel and Penot, 1984) respectively. These definitions were motivated by the need to extend differential calculus beyond smooth or convex/concave functions. When f : E ^ R is merely Lip-schitz around x, the said derivatives are sublinear in the direction d, and f ^(x; ■) < f 0(x; ■). Therefore, using M’s apparatus (14), one may define associated subdifferentials
df0(x) := {x* : (x*, d) < f0(x; d), Vd} , df^(x) := (x* : (x*,d) < f^(x; d), Vd}
to obtain w*-compact sets df^(x) C d0f (x) that figure in non-smooth differential calculus and reproduce (15), (16) via (13):
f0 (x; d) = max {(x*, d) : x* <G df0 (x)} , f ^ (x; d) = max ((x*, d) : x* <G df ^ (x)}
The connections f0(x; ■) ^ df0(x), f^(x; ■) ^ df^(x) are characteristic of the role played by convexity - and notably by Minkowski’s constructions - in modern
analysis; they are instrumental in the passage from linear/ smooth/ uni-valued relations to corresponding nonlinear/ nonsmooth/ multi-valued objects. In short, M was first to build bridges between convex functions and sets, between convex geometry and analysis.
Functions (15), (16) provide conical (sublinear) approximations of / at the point x of reference. Basic in this regard is the conical approximation of a convex body C at any boundary point x. Specifically, M defines the (shifted) tangent cone (Projektionsraum GA p. 161)
x + TC (x) := cl {x + R++ (intC — x)} ,
and he identifies it with the intersection of all supporting half-planes passing through x; that is,
x + TC (x) = {e G E : (x*, e) < 8*(x*) whenever 8*(x*) = (x*, x)} .
A full circle of constructions closes here, namely: If C = {y < 1} for a positive, sublinear y, and x G bdC = {y = 1} , then
o, ^ -o, ^ l(x +r') - 7(x) ,, ^
7 (x,') = 7 (x,') = 1™ ----------------------=: 7 (x,')
r—>0+ r
and TC(x) = {y'(x, ■) < 0}. Vectors x* for which 8*(x*) = (x*,x) are called normals to C at x; they constitute a closed convex cone NC (x), (composed of exactly those points x* — x such that x* has x as its best approximation in C.) M classifies boundary points of C according to the dimension of their normal cones. In particular, when intNC (x) is nonempty, he declares x a corner point (Eckpunkt). Such points, already encountered in connections with polytopes, are extreme and at most countably many.
A half-space {a < 0} , containing C, is called extreme if for any two different, bounding half-spaces {aj < 0} D C and positive numbers rj, i = 1,2, it holds that a = ri ai +r2a2. M shows in GA p. 166 that (6) obtains with a polytope P delineated only by extreme half-spaces of C. Preceding this notion, of course, is the more elementary concept of separation:
8. Separation, Polarity and Duality
M introduced the concept of a supporting hyperplane (with prescribed outward normal) to a closed set (GA p. 106), and shows - as a preliminary for later results -that every boundary point of a polytope (5) admits a supporting hyperplane. Any polytope is fully characterized by a finite family of supporting planes (GA p. 137). M went on to demonstrate, in two different ways, the following
Theorem (on supporting hyperplanes GA p.139). Through every boundary point of a convex body there passes a supporting hyperplane. □
M’s first proof uses the already obtained result for polytopes by invoking the twosided approximation (6). His second demonstration is distinctly modern in flavor, employing the orthogonal projection PC(x) := argmincGC ||x — c|| , shown to be attained. If x G C, then the hyperplane through x, having unit normal vector directed outwards along x — PC (x), does not intersect C. (This argument carries verbatim
over to Hilbert spaces.) When x € bdC, he supposes without loss of generality that 0 € intC and notes that x € rC for any r € (0,1). Now M replaces C by rC, repeats the preceding argument, and finally he let r f 1 to achieve the desired result. M proceeded to prove a first
Theorem (on separation, geometric form). Two convex bodies C, C' with C 0 intC' = 0, can be separated by a hyperplane.24 □
For the argument he again used compactness, continuity, and strict convexity as instruments. The proof goes broadly as follows: First assume C 0 C' = 0. The minimum distance infcGC,c'GC' ||c — c'|| is then positive and uniquely realized by two points c € C, c' € C' .In arguing this his use of closure and boundedness (i.e. compactness) has become standard, but was at that time quite modern. A plane with unit normal vector (c — c')/ ||c — c'|| , passing through any point strictly between c, c', will ensure separation. When C, C' have boundary points in common, he assumes 0 C, replaces C by rC, 0 < r < 1, and proceeds as described above by letting r f 1.
In GZ §18 M calls a convex body everywhere convex (uüberall convex) iff every supporting hyperplane touches merely in one point. Evidently, this amounts to strict convexity. Prime examples include lower level-sets C = {x : Q(x) < r} , r > 0, of positive quadratic forms Q; see GZ §49.25 Geometric considerations, akin to the classical inversion with respect to a circle (Hartshorne, 1997), lead M (GA p. 146) to call the lower level set C0 := {x° : S* (x0) < 1} the polar of C. Thus x0 € C0 iff x0, x < 1 for all x C. The symmetry of this relation he uses to show that for a convex body C with 0 € intC it holds that C00 := (C0)0 = C. This result is a first version of the so-called bipolar theorem: For any set C C E one has C00 = cl(conv(CU0)). Included, as special case, is the bipolar cone theorem, namely: for any set K C R one has (K-)- = cl(conv(R+ K)), see (9).
Polarity now serves particularly well in the study of so-called convex processes, these being set-valued maps whose graphs are closed convex cones (Borwein and Lewis, 2000), (Borwein, 1983). They provide a unifying format for linear maps, convex cones, and linear programming. M emphasizes the importance of duality by pointing out that the polar of a polytope P, containing 0 as interior point, is a set o} the same sort; see GA p. 146-147. Specifically: Suppose P = niGl {x : (xjo, x) < 1} for some finite set I. Then P0 = conv {xj0} ^.
The fundamental significance of M’s separation theorem came later to sharp light in functional analysis - on infinite-dimensional vector spaces - with various formulations of Hahn, Banach, Ascoli and Mazur (Dieudonne, 1981).26 Important
24 For an extension see Theorem 2.39 in (Rockafellar and Wets, 1998).
25 Another instance comes via Minkowski’s theorem on mixed volumes: Let
Cj C E be a convex body and rj > 0 for j = 1,...,n. Then volY^, j rj Cj = ^2v(Cj1 ,...,Cjn )rj1 ...rjn, the sum extending independently over all j1 ,...,jn. □ (The polynomial coefficients are called mixed volumes; see (Ewald, 1996).) In GA, Chap. XIII M takes two non-homothetic convex bodies C1, C2 in R3 and develops vol(C1 + rC2) as a polynomial of degree 3. By the Brunn-Minkowski theorem 0 < r ^ vol(C1 + rC2)1/3 is strictly concave. This yields an isoperimetric inequality which, for the special instance vol(C1) = 4nR3/3, C2 = unit ball, tells that, among convex bodies, for given volume the sphere has minimal surface.
26 (Holmes, 1975) contains ten different but equivalent formulations.
in such spaces are algebraic and topological versions of the corresponding so-called Hahn-Banach theorem, requiring that either the algebraic interior (4) or topological interior of some ground set be nonempty; see (Lassonde, 1988) and references therein.
Often, the said theorem comes in analytic (as opposed to the above geometric) form, saying that if a linear function on a linear subspace is bounded above by a sublinear function, then the first can be extended to the whole space while preserving the same upper bound. However, after representing the linear function by its graph and the sublinear function S by its epigraph epiS := {(e,r) G E x R : S(e) < r} , one may separate those two sets to have the desired linear extension. Thus, in essence, M’s geometric form remains as potent as the analytical versions. Also, in infinite dimensions things are turned somewhat around in that gauges and support functions become crucial for defining and understanding the (locally convex) topologies.
We shall not review this but rather return to (13). We insert there any convex function / : E ^ RU {+^} which is proper (i.e. / attains finite values somewhere) and lower semicontinuous (or closed for short; i.e. lim inf/(x') > /(x), Vx) to produce the so-called Fenchel convex conjugate
/*(x*) := sup {(x*,x) — /(x) : x G E}
The operator / ^ /* thus associates to a proper closed convex function / on E another proper closed convex function /* defined on the dual space E*. Moreover, using M’s separation theorem, Fenchel (1949) showed that /** := (/*)* = / (and thereby that convex conjugation is bijective). Convex conjugacy thus gives a invo-lutory dual description of proper closed functions. M’s merit was to initiate that description for the special case of proper closed sets (that is, for extended indicators) - as done in (13), (14). Various forms of duality, say of Langrangian or Fenchel type, are intimately related to convex conjugates. These manifestations of convexity have become central in theory as well as in computation (Borwein and Lewis, 2000), (Hiriart-Urruty and Lemarechal, 1993).
9. Concluding Remarks
To record and appreciate all contributions of M that invoke convexity, in one form or another, the ambitious historian must follow his impressive routes ¿from number theory to physics, from quadratic forms to relativity theory. Most readers, including us, lack sufficient preparation, motivation, or time to fully undertake such a long, all-comprising journey.27 It is, however, certainly well worth quite a while to see what and how much M contributed to modern analysis. His studies, often motivated by number theoretic issues, show a rich interplay between analytic, geometric and topological ideas - organized or culminating around the concept of
27 Hancock (1939) finds that M ” as an expositor was very poor.” We disagree out of three reasons. First, for fairness one should bear in mind that linear algebra and topology, both masterly used by M, were in their infancy around 1895 (and still not quite ’’modern” in Hancock (Hancock,1939)). Second, since M was a pioneer, offering manifold novelties, one can hardly expect didatics to keep full pace with his contributions. Third,, M died ”im Vollbesitz seiner Lebenskraft, aus der Mitte freudigstens Wirkens, von der Hohe seines wissenschaftlichen Schaffens,” (GA vol. I, p.V), before some works in GA had found their final form.
convex sets. This was a major novelty in number theory. It took however, about thirty years before convexity made its way into game theory (Ville, 1938)28 and functional analysis (Ascoli, 1932), ( Mazur, 1933). Two more decades passed before it became central in optimization theory (Fenchel, 1951). And only recently has the same interplay come to full fruition in nonsmooth (variational) analysis (Clarke, 1983), (Rockafellar and Wets, 1998). In these subjects, beginning with (Minkowski, (1896) ), the geometric nature of various issues have been unifying; see for example (Holmes, 1975).
Instrumental for all this development were M’s separation theorem and two classes of convex functions: gauges (12) and supports (13) - both sublinear, both introduced by him. While these constitute the most important special case of convex functions, the general concept came first with Jensen (1906). Later, the unifying epigraphical, ’’geometric” perspective of Fenchel (1951) brought convex sets and functions on equal footing. His optic made it possible to ’identify” tangent planes (cones) with generalized (directional) derivatives.
Interesting in this regard is that two main, early users of convex analysis - noncooperative game theory (Nash 1951) and mathematical economics (Debreu 1959) -let convexity substitute for uni-valuedness and smoothness. Equally interesting, in those fields, is the lapse of almost two hundred years - and the role of convexity in fixed point theorems - needed to consolidate the first, pioneering economic insights of A. Smith (1776) and A. Cournot (1838).
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