Harmonic Analysis of Translation Invariant Valuations

By Semyon Alesker, A. Bernig, Franz E. Schuster. 2010

104 citations recorded by the source corpus.

The decomposition of the space of continuous and translation-invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger-type theorem for continuous translation-invariant and SO(n)-equivariant tensor valuations is also given. As an application, symmetry properties of rigid-motion invariant and homogeneous bivaluations are established and then used to prove new inequalities of Brunn–Minkowski type for convex body valued valuations.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/eb8b5dd5dff88cb761b067b0c1c4118aa7cb23c4

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, unknown): “, [4]–[7], [12], [37]–[39], [47]–[51], and [76]–[77]).”
The module of unitarily invariant area measures (2012, methodology): “The decomposition of Valk,C under the action of SO(2n) into isotypical components was determined in [15].”
Valuations and Surface Area Measures (2014, unknown): “[2–6, 9, 13–15, 20–25, 34, 39–41]).”
The log-Brunn-Minkowski inequality (2012, background): “If K and L are origin-symmetric convex bodies in the plane, then for all real λ ∈ [0, 1], V ((1 − λ)·K +o λ·L) ≥ V (K )1−λV (L).”
Minkowski valuations and generalized valuations (2015, methodology): “5 ([10]) For 0 ≤ i ≤ n, the spaces Val∞i , Vali, and Val i are multiplicity free under the action of SO(n).”
Kinematic formulas for tensor valuations (2014, methodology): “The decomposition of the SO(n)-module Valk has recently been obtained in [13].”