Characterizing the dual mixed volume via additive functionals

Abstract

Integral representations are obtained of positive additive functionals on finite products of the space of continuous functions (or of bounded Borel functions) on a compact Hausdorff space. These are shown to yield characterizations of the dual mixed volume, the fundamental concept in the dual Brunn-Minkowski theory. The characterizations are shown to be best possible in the sense that none of the assumptions can be omitted. The results obtained are in the spirit of a similar characterization of the mixed volume in the classical Brunn-Minkowski theory, obtained recently by Milman and Schneider, but the methods employed are completely different.

Characterization of dual mixed volumes via polymeasures

“…The result above includes and extends the main results in [10]. To see this, note that, as the authors mention in that paper, additive functions taking values in [0, ∞) are always positively homogenenous.…”

“…In Theorem 3.4 below we give conditions which do suffice. Condition (1) below appeared already in [10]. Condition (2) is new.…”

“…Following [19] and [10], our approach is to characterize the dual mixed volume based on its functional properties.…”

“…In the papers [10,19], the authors study different characterizations of the mixed volume and the dual mixed volume by their respective functional properties.…”

“…One of these conditions is stated in terms of the polynomial associated to the dual mixed volume, which turns out to be the n-dimensional volume. As a byproduct we also reprove the functional characterization of the dual mixed volume that appeared in [10].…”