Log-Concavity Properties of Minkowski Valuations

Minkowski valuations and generalized valuations

“…Theorem 3 (a) for i = 1 was recently proved by Alesker, see [11,Appendix]. Theorem 3 (b) for i = n − 1 follows from a classical result of McMullen [48].…”

“…In the final section, we need a generalization of (2.24) that can be deduced from [25, Theorem 4.3] and was independently proved in [11]: For every j ∈ {2, . .…”

“…Characterizations of Minkowski valuations, in particular, earlier versions of Theorem 2, have had far reaching implications for isoperimetric type inequalities (see, e.g., [2,11,32,[44][45][46]61]). Motivated by a recent important Crofton type formula for the identity map of Goodey and Weil [25], we show in the final section of this paper how Theorem 2 can be applied to obtain a general Crofton formula for continuous Minkowski valuations which generalizes the result from [25] and an earlier result of this type from [62].…”

Affine vs. Euclidean isoperimetric inequalities

“…This is in stark contrast to the case of Minkowski valuations intertwining only rigid motions which form an infinite dimensional cone. Nonetheless, also here substantial inroads towards a complete classification have been made (see [26,54,57,59,60]), making it possible to extend affine inequalities for the projection body operator by Lutwak [38] to a much larger class of Minkowski valuations (see [1,4,6,49,56,57]). This raised the natural problem whether the Petty projection inequality also holds in greater generality.…”