On P. McMullen's Conjecture on Translation Invariant Valuations

✍ Scribed by S Alesker

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 199 KB
• Volume: 155
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2000.1918

✦ Synopsis

dedicated to professor vitali d. milman on the occasion of his 60th birthday

1. Introduction

Let R n be the n-dimensional linear space. Let K n denote the family of all convex compact subsets of R n .

Definition 1.1. A scalar valued function

is called a valuation if for every two convex compact sets K 1 , K 2 such that their union is also convex one has

The K n is equipped naturally with the Hausdorff metric and it becomes a locally compact space. In this work we will study only translationinvariant valuations continuous with respect to the Hausdorff metric. Let us recall that the valuation , is called translation invariant if ,(K+x)=,(K) for every convex compact set K # K n and for every vector x # R n . Clearly the space of all such valuations is a linear space.

The basic example of valuations of this type is the mixed volume of K taken j times with some fixed convex compact sets A 1 , ..., A n& j , i.e., ,(K)=V(K[ j], A 1 , ..., A n& j ) (see [Sch1] for definitions and further details).

The classical result in the valuation theory is the Hadwiger characterization of isometry-invariant continuous valuations [H] (see also [K] for a simpler proof