Integrals of Smooth and Analytic Functions over Minkowski's Sums of Convex Sets

Summary

1. Introduction and Statement of Main Results

Let K = (K₁, K₂ , … , K_s) be an s-tuple of compact convex subsets of ℝⁿ . For any continuous function F : ℝⁿ → 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 (ℝⁿ) the Frechet space of r times differentiable functions on ℝⁿ with the topology of the uniform convergence on compact sets in ℝⁿ of all partial derivatives up to the order r (1 ≤ r ≤ ∞). The main results of this work are Theorems 1 and 3 below.