Improved log-concavity for rotationally invariant measures of symmetric convex sets

Rigidity of Invariant Convex Sets in Symmetric Spaces

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

Gaussian measures of dilatations of convex symmetric sets

Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures

Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, Stoch. Proc. Appl., 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-tr...

Convex, Rotationally Symmetric Liquid Bridges between Spheres

A liquid bridge between two balls will have a free surface which has constant mean curvature, and the angles of contact between the free surface and the fixed surfaces of the balls will be constant (although there might be two different contact angles: one for each ball). If we consider rotationally symmetric bridges, then the free surface must be a Delaunay surface, which may be classified as ...

A strong log-concavity property for measures on Boolean algebras

We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner [15]; a new proof of a theorem of Liggett [9] stating that ultra-log-concavity of sequences is preserved by convolutions; and some progres...