In this paper we give refinements of some convex and log-convex moment inequalities of the first and second order using a special kind of positive definite form. An open problem concerning eight parameter refinement of second order is also stated with some applications in Information Theory.

We present algorithms for computing all placements of two and three fingers that cage a given polygonal object with n edges in the plane. A polygon is caged when it is impossible to take the polygon to infinity without penetrating one of the fingers. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging placements of two disc fingers in O(...

Considering the space of closed subsets of R, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)nR, we prove that the only minimal subsystems are the fixed points {∅} and {R}. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set Y ⊂ R such that for every convex set C ⊂ R of volume one, the cardinality of C ∩Y is ...

We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O (n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.

This paper describes a log-linear modeling framework suitable for large-scale speech recognition tasks. We introduce modifications to our training procedure that are required for extending our previous work on log-linear models to larger tasks. We give a detailed description of the training procedure with a focus on aspects that impact computational efficiency. The performance of our approach i...

For every polynomial time algorithm which gives an upper bound vol(K) and a lower bound vol(K) for the volume of a convex set K = R d, the ratio vo~(K)/vo__!l(K) is at least (cd/log d) a for some convex set K = R d.

We give a simple O(n log n) algorithm to compute the convex hull of the (possibly Θ(n)) intersection points in an arrangement of n line segments in the plane. We also show an arrangement of dn planes in d-dimensions whose arrangement has Θ(nd−1) intersection points on the convex hull.

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.