Semyon Alesker

TL;DR: The main result of as mentioned in this paper is the description of translation invariant continuous valuations on convex sets, which provides an affirmative solution of P McMullen's conjecture, and in a stronger form.

TL;DR: The notion of valuation on convex sets can be considered as a generalization of the notion of measure, which is deflned only on the class of convex compact sets as mentioned in this paper.

TL;DR: In this paper, the structure of the space of translation invariant continuous valuations on convex sets has been characterized and the hard Lefschetz theorem has been shown to imply new integral geometric formulas for real submanifolds in Hermitian spaces.

TL;DR: In this paper, a canonical structure of a commutative associative filtered algebra with unit is introduced on the space of polynomial smooth valuations, and its properties are studied.

TL;DR: A characterization of continuous isometry covariant valuations on convex sets is presented and the main result generalizes previous results of Hadwiger and Hadwinger and Schneider.