Interaction of Order and Convexity

S. S. Kutateladze

An expanded version of the talk is available in PDF and PPT.

0.1 A measure μ linearly majorizes or dominates a measure ν provided that to each decomposition of SN-1 into finitely many disjoint Borel sets U1,...,Um there are measures μ1,...,μm with sum μ such that every difference μk - ν|Uk annihilates all restrictions to SN-1 of linear functionals over ℝN. In symbols, we write μ » N ν. It is well known that
SN-1 p dμ ≥ ∫SN-1 p dν
for every sublinear functional p on ℝN iff μ » N ν.
0.2convex figure is a compact convex set. A convex body is a solid convex figure. The Minkowski duality identifies a convex figure S in ℝN with its support function S(z):=sup{(x,z) | x ∈ S} for z ∈ ℝN. Considering the members of ℝN as singletons, we assume that ℝN lies in the set VN of all compact convex subsets of ℝN.
0.3 The Minkowski duality takes VN into a cone in the space C(SN-1) of continuous functions on the Euclidean unit sphere SN-1, the boundary of the unit ball. This yields the so-called Minkowski structure on VN. Addition of the support functions of convex figures amounts to taking their algebraic sum, also called the Minkowski addition. It is worth observing that the linear span [VN] of  VN is dense in C(SN-1), bears a natural structure of a vector lattice, and is usually referred to as the space of convex sets. The study of this space stems from the pioneering breakthrough of Alexandrov in 1937 and the further insights of Radström, Hörmander, and Pinsker.
0.4 The talk will discuss the details of the relevant functional analytical techniques and applications to the extremal problems of convex geometry.


[1] Reshetnyak Yu. G. (1954) On the Length and Swerve of a Curve and the Area of a Surface (Ph. D. Thesis). Leningrad State University [in Russian].
[2] Alexandrov A. D. Selected Scientific Papers. London etc.: Gordon and Breach (1996).
[3] Radström H. (1952) An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc., 3:1, 165–169.
[4] Hörmander L. (1955) Sur la fonction d'appui des ensembles convexes dans une espace lokalement convexe. Arkiv för Math., 3:2, 180–186 [in French].
[5] Pinsker A. G. (1966) Some classes of the spaces of convex sets and their completions// In: Some Classes of Semi-Ordered Spaces. Leningrad: Leningrad State University, 13–18 [in Russian].

April 26, 2007

The talk was delivered at the opening of the Third Russian–German Geometry Meeting Dedicated to the 95th Birthday of A. D. Alexandrov; St. Petersburg, June 18–23, 2007.

A complete version is available:
J. Appl. Indust. Math., 2007, V. 1, No. 4, 399–405
+ arXiv:0705.4124v1 [math.FA]

On June 17, 2002 at the PDMI
I.A. Taimanov and S.S. Kutateladze in the Euler Institute

File translated from TEX by TTH, version 3.77.
On 29 May 2007, 20:47.

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