Let A(S) be the arrangement formed by a set of n line segments S in the plane. A subset of arrangement vertices p1, p2, . . . , pk is called a convex k-gon of A(S) if (p1, p2, . . . , pk) forms a convex polygon and each of its sides, namely, (pi, pi+1) is part of an input segment. We want to count the number of distinct convex k-gons in the arrangement A(S), of which there can be Θ(n) in the wo...

We prove that the convex least squares estimator (LSE) attains a n-1/2 pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent test...

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4. Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the rst algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms fo...

The largest empty circle (LEC) problem is defined on a set P and consists of finding the largest circle that contains no points in P and is also centered inside the convex hull of P . The LEC is always centered at either a vertex on the Voronoi diagram for P or on an intersection between a Voronoi edge and the convex hull of P . Thus, finding the LEC consists of constructing a Voronoi diagram a...

We develop a parallel theory to that concerning the concept of integral mean value of a function, by replacing the additive framework with a multiplicative one. Particularly, we prove results which are multiplicative analogues of the Jensen and Hermite-Hadamard inequalities. 1. Introduction Many results in Real Analysis exploits the property of convexity of the subintervals I of R; (A) x; y 2 I...

In this paper, we establish some Hermite-Hadamard type inequalities for function whose n-th derivatives are logarithmically convex by using Riemann-Liouville integral operator.

We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman’s reverse Brunn-M...

We consider the problem of optimizing an approximately convex function over a bounded convex set in Rn using only function evaluations. The problem is reduced to sampling from an approximately log-concave distribution using the Hit-and-Run method, which is shown to have the same O∗ complexity as sampling from log-concave distributions. In addition to extend the analysis for log-concave distribu...