Integrals of Smooth and Analytic Functions over Minkowski's Sums of Convex Sets
Semyon Alesker · 1999
1. Introduction and Statement of Main Results Let K = ( K 1 , K 2 , … , K s ) be an s-tuple of compact convex subsets of ℝ n . For any continuous function F : ℝ n → ℂ, consider the function This defines an operator M K , which we will call a Minkowski operator. Denote by A ( ℂ n ) the Frechet space of entire functions in n variables with the usual topology of the uniform convergence on compact sets in ℂ n , and C r ( ℝ n ) the Frechet space of r times differentiable functions on ℝ n with the topology of the uniform convergence on compact sets in ℝ n of all partial derivatives up to the order r (1 ≤ r ≤ ∞). The main results of this work are Theorems 1 and 3 below.