0.1
A measure μ
linearly majorizes or
dominates
a measure ν provided that to each decomposition of
S
_{N1} into finitely many disjoint Borel sets U
_{1},...,U
_{m}
there are measures μ
_{1},...,μ
_{m} with sum
μ
such that every difference μ
_{k}  ν
_{Uk} annihilates
all restrictions to S
_{N1} of linear functionals over
ℝ
^{N}. In symbols, we write μ »
_{ℝN}
ν.
It is well known that

∫_{SN1} p dμ ≥
∫_{SN1} p dν




for every sublinear
functional p
on ℝ
^{N} iff μ »
_{ℝN}
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
_{N1}) of continuous functions on the Euclidean unit sphere
S
_{N1}, the boundary of the unit ball.
This yields the socalled
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
_{N1}), 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.