Dmitry Faifman (Tel Aviv University)
Title: Nash and Whitney problems in convex valuation theory
Abstract: A (convex, smooth) valuation is a finitely additive measure on convex bodies, satisfying a smoothness condition; many interesting objects in convex and differential geometry are in fact valuations.
Assume that a collection of valuations is given on a family S of subspaces of R^n . Are they the restrictions of a single valuation? Clearly, compatibility of the given data on intersections is a necessary condition. Is it sufficient?
We will discuss several geometrically distinct instances of this problem, whence it acquires distinct flavors.
When S is the whole k-grassmannian, and the valuations j-homogeneous, we will see that the condition is sufficient, provided k-j>1. This can be seen as a dimensional localization of the transition from densities to valuations.
In another setting where S consists of pairwise non-intersecting subspaces, we again establish a positive answer. As a corollary, we will deduce a Nash embedding theorem for smooth valuations on manifolds.
Finally, we will consider the setting of finite generic families of subspaces, giving rise to a surprising extension phenomenon.
Based on a joint work with Georg Hofstaetter.
CRM, Universit茅 de Montr茅al, Pavillon Andr茅-Aisenstadt, room 5340, and by Zoom (see link below)
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