Convex valuations invariant under the Lorentz group

Abstract

We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.where a valuation φ is called even (resp. odd) if φ(−K) = φ(K) (resp. φ(−K) = −φ(K)) for any K.It is easy to see that V al 0 (R n ) is spanned by the Euler characteristic, i.e. valuation which is equal to 1 on any convex compact set. By a theorem of Hadwiger [14], V al n (R n ) is spanned by the Lebesgue measure.We denote by V al(R n ) SO + (n−1,1) the subspace of SO + (n − 1, 1)-invariant valuations, and similarly for subspaces of given parity and homogeneity. Mc-Mullen's decomposition (1) immediately impliesOur first main result classifies odd SO + (n − 1, 1)-invariant valuations.Theorem 1.1. For 0 ≤ k ≤ n, k = n − 1, V al odd k (R n ) SO + (n−1,1) = 0. For k = n − 1, dim V al odd k (R n ) SO + (n−1,1) = 1, n ≥ 3 2, n = 2as claimed. Q.E.D.

Kinematic formulae for tensorial curvature measures

“…Various other groups of isometries, also in Riemannian isotropic spaces, have been studied in recent years. Major progress has been made, for instance, in Hermitian integral geometry (in curved spaces), where the interplay between global and local results turned out to be crucial (see [13,14,25,26,79,80,76] and the survey [11]), but various other group actions have been studied successfully as well (see [4,8,9,12,17,18,19,23]).…”

Minkowski Additive Operators Under Volume Constraints

“…Let Φ ∈ MVal and w ∈ R n . (i) follows immediately from (3) and (4). The proof of (ii) is a consequence of (4) and the homogeneity of Φ.…”

“…Real-valued valuations were probably first used by Dehn for his solution of the third Hilbert problem. The reader interested in the state of the art of the theory of real-valued valuations is referred to the valuable surveys [2,6,15,26,29,40,41] and [50,Chapter 6], and to [3,4,7,8,35] for the most recent results. Nowadays, apart from real-valued and Minkowski valuations, other valuations, namely, for other abelian semigroups (A, +) than the reals or (K n , +), have been studied, often motivated by their applications in material science and physics.…”

Crofton formulas and indefinite signature

“…The Lorentz group O(n−1, 1) was considered by S. Alesker and the author in [8], and the general signature was studied by A. Bernig and the author in [14]. There, the dimensions of the spaces of invariant valuations were computed, and a simple description was given in terms of their Klain sections.…”

“…In [9], Alesker and Bernstein have shown that the image of Cr is dense in Val + k (V ), equipped with the topology of uniform convergence on compact sets. In a later paper [8], Alesker and the author have shown that in fact, any valuation φ ∈ Val + k (V ) can be represented as Cr(µ), where µ ∈ M −∞ (AGr n−k (V )) tr is some translation-invariant distribution (generalized measure). Moreover, if one replaces the even continuous valuations with the somewhat larger class of even generalized valuations Val +,−∞ (V ) (in which the former is a dense subspace), one can extend Cr : M −∞ (AGr n−k (V )) tr → Val +,−∞ k (V ) as a surjection.…”