Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture

By Semyon Alesker. 2001

234 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0d4af56cca187ae84bf4f3c081489c3e464525a3

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Valuations on Polytopes Containing the Origin in Their Interiors (2002, background): “Recently, Alesker [ 6 ] has given a complete classification of continuous, translation invariant valuations on the space of convex bodies thereby confirming a twenty year old conjecture by McMullen [19].”
Projection bodies in complex vector spaces (2011, background): “e-dimensional real vector space of dimension n. The Banach space of real-valued, translation invariant, continuous valuations on V is denoted by Val. It has been studied intensively, see for instance [1, 2, 20, 38]. A basic structure result for Val is the following direct sum decomposition due to Mc”
The product on smooth and generalized valuations (2009, methodology): “H◦ L|ˆP\N −1 : P\M → P\M (x,y,[ξ: η]) → (x+dξhA,y+dηhB,[ξ: η]). Since N(K) annihilates vertical forms, the same holds true for H∗Nˆ(K). Let us consider the second term. We have a commuting diagram [0,1] ×N1 H˜ 1 / qX◦τ1 M1 PX HA /P X where the vertical map on the right is given by (x,y,[ξ: 0]) → ”
Valuations on convex functions and convex sets and Monge–Ampère operators (2017, background): “ew words on the proof of this result. The map T commutes with the natural actions on both spaces of group GL(V) of invertible linear transformations on V. Hence by the author’s irredicibility theorem [2] it suﬃces to show that Im(T) intersects non-trivially each subspace of Val(V) consisting of valu”
A fourier-type transform on translation-invariant valuations on convex sets (2007, methodology): “The construction in the odd case is quite involved and uses various characterization theorems on valuations, in particular Klain–Schneider characterization of simple valuations [31], [41] (see Theorem 2.3.6 below), the Irreducibility Theorem [ 3 ] (see Theorem 2.3.7 below), and also some additional ”
Crofton Measures and Minkowski Valuations (2010, background): “In 2001, Alesker [2] has given a complete description of continuous translation invariant valuations on convex bodies thereby confirming, in a much stronger form, a conjecture by McMullen.”
Projection bodies and valuations (2002, background): “In recent years, many new results on real and tensor valued valuations have been obtained (see, for example, [1]‐[3], [13]‐[16], [19], [30], and Klain and Rota’s book [17]), including Alesker’s proof [ 4 ] of McMullen’s twenty year old conjecture on the classification of translation invariant valuat”
Rotation equivariant Minkowski valuations (2012, background): “In recent years, many new results on real and body valued valuations have been obtained, see [1, 2, 10, 11, 13, 14, 15, 16, 17, 27].”
A Hadwiger-Type Theorem for the Special Unitary Group (2008, background): “This is a consequence of Alesker’s irreducibility theorem [2] and the kernel theorem of [12].”