The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures

Abstract

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge-Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.

The Hadwiger theorem on convex functions, IV: The Klain approach

“…This extension is called the jth functional intrinsic volume V n j,ζ with density ζ. Explicit representations for functional intrinsic volumes were obtained in [14] using functional Cauchy-Kubota formulas (see Theorem 5.3 below) and in [15] using a new family of mixed Monge-Ampère measures. Note that V n 0,ζ (u) is a constant for ζ ∈ D n 0 , independent of u ∈ Conv sc (R n ).…”