We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of volume a convex body and its polar. confirm for hyperplane sections or projections $\ell_p$-balls Hanner polytopes.

We discuss connections between certain well-known open problems related to the uniform measure on a high-dimensional convex body. In particular, we show that the “thin shell conjecture” implies the “hyperplane conjecture”. This extends a result by K. Ball, according to which the stronger “spectral gap conjecture” implies the “hyperplane conjecture”.

We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky’s fundamental results on the number of regions.

For any finite, real reflection group W , we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs in [4] using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by W .

This note establishes a connection between Solomon’s descent algebras and the theory of hyperplane arrangements. It is shown that card-shuffling measures on Coxeter groups, originally defined in terms of descent algebras, have an elegant combinatorial description in terms of random walk on the chambers of hyperplane arrangements. As a corollary, a positivity conjecture of Fulman is proved.

This note establishes a connection between Solomon's descent algebras and the theory of hyperplane arrangements. It is shown that card-shu ing measures on Coxeter groups, originally de ned in terms of descent algebras, have an elegant combinatorial description in terms of randomwalk on the chambers of hyperplane arrangements. As a corollary, a positivity conjecture of Fulman is proved. 2

The number of external facets of a simple arrangement depends on its combinatorial type. A computation framework for counting the number of external facets is introduced and improved by exploiting the combinatorial structure of the set of sign vectors of the cells of the arrangement. 1 Background and introduction n hyperplanes in dimension d form a hyperplane arrangement. An hyperplane arrangem...