Log-concavity properties of Minkowski valuations

Minkowski Valuations

Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of Lp-Minkowski valuations. 2000 AMS subject c...

Rotation Equivariant Minkowski Valuations

The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivial operator with these properties. MSC 2000: 52B45, 52A20

Log-Concavity and Related Properties of the Cycle Index Polynomials

Let An denote the n-th cycle index polynomial, in the variables Xj , for the symmetric group on n letters. We show that if the variables Xj are assigned nonnegative real values which are logconcave, then the resulting quantities An satisfy the two inequalities An−1An+1 ≤ An ≤ ( n+1 n ) An−1An+1. This implies that the coefficients of the formal power series exp(g(u)) are log-concave whenever tho...

Log - Concavity and Related Properties of the Cycle Index

Matroids and log-concavity

We show that f -vectors of matroid complexes of realizable matroids are strictly log-concave. This was conjectured by Mason in 1972. Our proof uses the recent result by Huh and Katz who showed that the coefficients of the characteristic polynomial of a realizable matroid form a log-concave sequence. We also prove a statement on log-concavity of h-vectors which strengthens a result by Brown and ...