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Владикавказский математический журнал Июль сентябрь, 2002, Том 4, Выпуск 3
УДК 513.588
ONE FUNCTIONAL-ANALYTICAL IDEA BY ALEXANDROV IN CONVEX GEOMETRY
S. S. Kutateladze
The functional-analytical approach by A. D. Alexandrov is discussed to the Minkowski and Blaschke structures making the set of convex compact figures into a vector space. The resulting analytical possibilities are illustrated by the isoperimetric type problems of finding convex figures separated by current hyperplanes similar to the Urysohn and double bubble problems.
A. D. Alexandrov enriched convex geometry with the technique of functional analysis and measure theory (cf. [1]). The aim of this talk is to draw attention to some extra analytical possibilities in studying by his methods the isoperimetric type problems of the theory of convex surfaces of optimal location of several figures in the cells generated by a family of hyperplanes with prescribed normals. These are called problems with current polyhedra. They belong to the class of extremal problems with free boundary and are of profound interest due to various applied problems involving optimal location of figures. As examples we can list the «convex» versions and analogs of the «double bubble» problem and similar «soap» problems (cf. [2-5] and the references therein). In this talk we deal with some model examples illustrating several tricks for introducing extra inclusion constraints in the isoperimetric problems of the Urysohn and double bubble types. The results of this talk are partly announced in [6, 7].
It is well known that the classical Minkowski duality which identifies a convex figure y in M.N with its support function y(z) := sup{(a;, z) \ x € y} for z €E M.N. Considering the members of M.N as singletons, we assume that M.N lies in the set of all compact convex subsets Vjv of M.N. The Minkowski duality induces in Vjv the structure of a cone in the space C'(S'jv-i) of continuous functions on the Euclidean unit sphere S'jv-i, the boundary of the unit ball 3jv- This parametrization is the Minkowski structure. Addition of the support functions of convex figures amounts to passing to the algebraic sum of the latter, also called the Minkowski addition. It is worth observing that the linear span [Vjv] of the cone Vjv is dense in C'(S'jv-i).
The second parametrization, Blaschke structure, results from identifying the coset of translates {z + y | z €E M.N} of a convex body y, which is by definition a convex figure with nonempty interior, and the corresponding measure on the unit sphere which we call the surface area function of the coset of y and denote by //(y). The soundness of this parametrization rests on the celebrated Alexandrov Theorem of recovering a convex surface from its surface area function. Each surface area function is an Alexandrov measure. So we call a positive measure on the unit sphere which is supported by no great hypersphere and which annihilates singletons. The last property of a measure is referred to as translation invariance in the theory of convex surfaces. Thus, each Alexandrov measure is a translation-invariant additive functional over the cone Vjv- The cone of positive translation-invariant
© 2002 Kutateladze S. S.
measures in the dual C'(Sn-i) of C(Sn-i) is denoted by An- We now agree on some preliminaries.
Given y, t) £ Vjv, we let the record y = Rjvt) mean that y and tj are equal up to translation or, in other words, are translates of one another. We may say that = rjv is the equivalence associated with the preorder ^ rjv on Vjv symbolizing the possibility of inserting one figure into the other by translation. Arrange the factor set Vjv/№.N which consists of the cosets of translates of the members of Vjv- Clearly, Vn/№n is a cone in the factor space [Vjv]/®^ of the vector space [Vjv] by the subspace M.N.
There is a natural bijection between Vn/№N and An- Namely, we identify the coset of singletons with the zero measure. To the straight line segment with endpoints x and y, we assign the measure
\x ~ y\{e(x-y)/\x-y\ + £(y-x)/\x-y\):
where | • | stands for the Euclidean norm and the symbol for z £ Sn-i stands for the Dirac measure supported at z. If the dimension of the affine span Aff(y) of a representative y of a coset in Vjv/№.N is greater than unity, then we assume that Aff (y) is a subspace of M.N and identify this class with the surface area function of y in Aff(y) which is some measure on Sn-i H Aff(y) in this event. Extending the measure by zero to a measure on Sn-i, we obtain the member of An that we assign to the coset of all translates of y. The fact that this correspondence is one-to-one follows easily from the Alexandrov Theorem.
The vector space structure on the set of regular Borel measures induces in An and, hence, in Vjv/M.N the structure of a cone or, strictly speaking, the structure of a commutative M+ -operator semigroup with cancellation. This structure on Vn/№n is called the Blaschke structure. Note that the sum of the surface area functions of y and tj generates a unique class y#tj which is referred to as the Blaschke sum of y and tj.
Let C(Sn-i)/№n stand for the factor space of C(Sn-i) by the subspace of all restrictions of linear functionals on M.N to Sn-i- Denote by [A \J the space An — An of translation-invariant measures. It is easy to see that \A \J is also the linear span of the set of Alexandrov measures. The spaces C(Sn-i)/№-n and \A \J are set in duality by the canonical bilinear form
</,/*> = 1 [ fd/i (/£ C(Sn-i)№N, V £ [AN]). iV JSn-!
For y £ Vn/№n and tj £ A \. the quantity (y, tj) coincides with the mixed volume Fi(ti,y). The space [A v J is usually furnished with the weak topology induced by the above indicated duality with C'(S'jv-i)/IRjv •
By the dual K* of a given cone K in a vector space X in duality with another vector space Y, we mean the set of all positive linear functionals on K; i. e., K* := {y £ Y \ (Vx £ K)(x,y) ^ 0}. Recall also that to a convex subset U of X and a point x in U there corresponds the cone
Us := Fd(U, x):={h(EX\ (3a ^ 0) x + ah £ U}
which is called the cone of feasible directions of U at x. Fortunately, description is available for all dual cones we need.
1. The dual A*n of An is the positive cone of C(Sn-i)/№n .
2. Let у Є An- Then the dual A*N ^ of the cone of feasible directions of Лдгп at у may be represented as follows
A*N-T = {ftA*N\(fJ) = 0}.
Assume that ц and v are positive measures on the sphere <Sjv-i- Say that ц is linearly stronger than v and write ц » жмь> if to each decomposition of v into the sum of finitely many positive terms v = v\ + ... + vm there exists a decomposition of ц into the sum of finitely many terms ц = + ... + fim such that ц\. — ь>к (Ж^ )* for all k = 1,... , m.
3. Let у and tj be convex figures. Then
(1) ju(y) - ju(o) <EV*N ^ ju(y) » KNn(t));
(2) If у ^ Rjvtj then ц( y) » Riv//(t));
(3) У ^ K2t) //(y) » R2//(t>).
4. Let у and t) be convex figures. Then
(1) If t) - у Є A\ y then t) =rjv y;
(2) If n{xi) - n{f) Є V;'v v then t) =rjv y.
In the sequel we never distinguish between a convex figure, the respective coset of translates in Vn/№n, and the corresponding measure in An-
It is worth noting that the volume F(y) := (y,y) of a convex figure у is a homogeneous polynomial of degree N with respect to the Minkowski structure. That is why to calculate the subdifferential of V(-) is an easy matter. The particular feature of the Minkowski structure is an intricate construction of the dual of the cone of compact convex sets whose description bases on the relation » rjv in the space of measures \A \ J.
5. External JJrysohn Problem. Among the convex figures, including y0 and having integral width fixed, find a convex body of greatest volume.
Object of Parametrization Minkowski’s Structure Blaschke’s Structure
cone of sets Vjv/mN An
dual cone V* VN A*
positive cone A* V An
typical linear Vl(iN, ') Vi(-,3 n)
functional (width) (area)
concave functional Vl!N{-) y(N-l)/N(.)
(power of volume)
simplest convex isoperimetric Urysohn’s
program problem problem
operator-type inclusion inequalities
constraint of figures on «curvatures»
Lagrange’s multiplier surface function
differential of volume at a point у
is proportional to vi(-,y)
Isoperimetric-type problems with subsidiary constraints on location of convex figures comprise in a sense a unique class of meaningful problems of mathematical programming which admits two essentially different parametrizations, the Minkowski and Blaschke structures. Their principal features are clearly seen from the table.
This table shows that the classical isoperimetric problem is not a convex program in the Minkowski structure for N ^ 3. In this event a necessary optimality condition leads to a solution only under extra regularity conditions. Whereas in the Blaschke structure this problem is a convex program whose optimality criterion reads: «Each solution is a ball.»
The task of choosing an appropriate parametrization for a wide class of problems is practically unstudied in general. In particular, those problems of geometry remain unsolved which combine constraints each of which is linear in one of the two vector structures on the set of convex figures. The simplest example of an unsolved «combined» problem is the internal isoperimetric problem in the space M.N for N ^ 3.
The above geometric facts make it reasonable to address the general problem of parametrizing the important classes of extremal problems of practical provenance.
In the sequel we use the following notations:
(y e Vn/M.n);
p:y^F0v-D/iV(y) (?eyljv).
The Minkowski inequality is thus paraphrased as (y, tj) ^ p(f)p(t}).
Let us illustrate the above by an «external» version of the Urysohn problem with an inclusion constraint «from below.»
6. Optimality Test. A feasible convex body y is a solution to Problem 5 if and only if there are a positive critical measure ft and a positive real a 6 M+ satisfying
(1) an{i-jy) » + n;
(2) V{y) + jffSN_i fd/i = aFi(3jv,y);
(3) f(z) = y0(z) for all z in the support of fi.
If, in particular, y0 = Jjv-i then the sought body is a spherical lens, that is, the intersection of two balls of the same radius; while the critical measure is the restriction of the surface area function of the ball of radius to the complement of the support
of the lens to S'jv-i- If yo = 31 and N = 3 then our result implies that we should seek a solution in the class of the so-called spindle-shaped constant-width surfaces of revolution.
Note also that, combining the above tricks, we may write down the Euler-Lagrange equations for a wide class of isoperimetric-type extremal problems. In particular events, these are reasonable to enpower with another technique of geometry and optimization. To illustrate this, we exhibit a rather typical example:
7. Urysohn Type Problem. Among convex figures of fixed thickness and integral width, find a convex body of greatest volume.
Recall that the thickness A(y) of a convex figure y is defined as follows:
A(y) := inf (y(z)+y(-z)).
z£S jv_i
Observe first that Problem 7 is stated as «convex on the wrong side.» However, applying the Minkowski symmetrization once, we see that a solution belongs to the class of centrally symmetric convex figures for which the restricted thickness may be rewritten as inclusion-type constraint.
8. Optimality Test. Let a positive measure ц and reals а, (3 € M+ satisfy the following conditions:
(1) afl($N) + M£Z0 + £-zo) ^ RNM(f) + Mi
(2) V(f) + ^/Sjv_1 ?Ф = aVi(3jv,y) + ^/3(y(zo) + y(-«o));
(3) f(z) = |A for all z in the support of ц.
Then a feasible convex body у is a solution to Problem 7.
Therefore, a convex figure a$N#j3}N-i of given integral width and thickness is optimal for Problem 7. In the case N = 3 , a solution belongs to the class of the so-called cheeseshaped constant-width surfaces of revolution.
9. Internal Urysohn Problem with a Current Hyperplane. Find two convex figures у and 5 lying in a given convex body yo, separated by a hyperplane with the unit outer normal zq, and having the greatest total volume off and 5 given the sum of their integral widths.
10. Optimality Test. A feasible pair of convex bodies у and 5 solves Problem 9 if and only if there are convex figures у and tj and positive reals a and (3 satisfying
(1) У =
(2) 5 = tj#«3iv;
(3) ju(y) ^ peZo, ju(o) ^ pe-Zo;
(4) f(z) = уо(г) for all z € supp(y) \ {zq};
(5) t)(z) = Уо(г) for all z € supp(y) \ zQ},
with supp(y) standing for the support of y, i.e. the support of the surface area measure ц{у) of у.
Note that some positive Blaschke-linear combination of the ball and a tetrahedron is proportional to the solution of the internal Urysohn problem in this tetrahedron (cf. [3]). In case N = 2 the Blaschke sum transforms as usual into the Minkowski sum. If we replace the condition on the integral width characteristic of the Urysohn problem (cp. [6]) by a constraint on the surface area or other mixed volumes of a more general shape then we come to possibly nonconvex programs for which a similar reasoning yields the necessary extremum conditions in general.
The above analysis together with the Schwarz symmetrization shows in particular that the solution of the «internal» double bubble problem within the ball in the class of the union of pairs of convex figures is given by an appropriate part of the union of some spherical caps.
Литература
1 Alexandrov A. D. Selected Works. Part 1: Selected Scientific Papers.—London etc.: Gordon and
Breach, 1996.—322 p.
2. Foisy J., Alfaro М., Brock J., Hodges N., and Zimba J. The standard double soap bubble in Ж2
uniquely minimizes perimeter // Pacific J. Math.—1993.—V. 159, No. 1.—P. 47-59.
3. Pogorelov A. V. Imbedding a ‘soap bubble’ into a tetrahedron // Math. Notes.—1994.—V. 56, No. 2.—
P. 824-826.
4. Hutchings М., Morgan Ritore М., and Ros A. Proof of the double bubble conjecture // Electron.
Res. Announc. Amer. Math. Soc.—2000.—V. 6, No. 6.—P. 45-49.
5. Urysohn P. S. Dependence between the average width and volume of convex bodies // Mat. Sb.—
1924,—V. 31, No. 3.—P. 477-486. [Russian]
6. Kutateiadze S. S. Parametrization of isoperimetric-type problems in convex geometry // Siberian Adv. Math.—1999.—V. 9, No. 3.—P. 115-131.
7. Kutateiadze S. S. On the isoperimetric type problems with current hyperplanes // Siberian Math. J. (to be published).—2002,—V. 43, No. 4,—P. 132-144. [Russian]
г. Новосибирск
Статья поступила 20 мая 2002 г.
