Interaction of Order and Convexity

0.1 A measure μ linearly majorizes or dominates a measure ν provided that to each decomposition of S_N-1 into finitely many disjoint Borel sets U₁,...,U_m there are measures μ₁,...,μ_m with sum μ such that every difference μ_k - ν|_Uk annihilates all restrictions to S_N-1 of linear functionals over ℝ^N. In symbols, we write μ » _ℝ^N ν. It is well known that

∫SN-1 p dμ ≥ ∫SN-1 p dν

for every sublinear functional p on ℝ^N iff μ » _ℝ^N ν. 0.2 A convex figure is a compact convex set. A convex body is a solid convex figure. The Minkowski duality identifies a convex figure S in ℝ^N with its support function S(z):=sup{(x,z) | x ∈ S} for z ∈ ℝ^N. Considering the members of ℝ^N as singletons, we assume that ℝ^N lies in the set V_N of all compact convex subsets of ℝ^N. 0.3 The Minkowski duality takes V_N into a cone in the space C(S_N-1) of continuous functions on the Euclidean unit sphere S_N-1, the boundary of the unit ball. This yields the so-called Minkowski structure on V_N. 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 [V_N] of  V_N is dense in C(S_N-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.