We present output sensitive techniques for the generalized reporting versions of the planar range maxima problem and the planar range convex hull problem. Our solutions are in the pointer machine model, for orthogonal range queries on a static point set. We solve the planar range maxima problem for two-sided, three-sided and four-sided queries. We achieve a query time of O(log n+c) using O(n) s...

let $omega_x$ be a bounded, circular and strictly convex domain of a banach space $x$ and $mathcal{h}(omega_x)$ denote the space of all holomorphic functions defined on $omega_x$. the growth space $mathcal{a}^omega(omega_x)$ is the space of all $finmathcal{h}(omega_x)$ for which $$|f(x)|leqslant c omega(r_{omega_x}(x)),quad xin omega_x,$$ for some constant $c>0$, whenever $r_{omega_x}$ is the m...

Let S be a finite set of n points in the plane in general position. A k-hole of S is a simple polygon with k vertices from S and no points of S in its interior. A simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. Moreover, a point set S is l-convex if there exists an l-convex polygonalization of S. Considering a typical Erdős-Szek...

Most online optimization algorithms focus on one of two things: performing well in adversarial settings by adapting to unknown data parameters (such as Lipschitz constants), typically achieving O( √ T ) regret, or performing well in stochastic settings where they can leverage some structure in the losses (such as strong convexity), typically achieving O(log(T )) regret. Algorithms that focus on...

We prove an isoperimetric inequality for uniformly log-concave measures and for the uniform measure on a uniformly convex body. These inequalities imply the log-Sobolev inequalities proved by Bobkov and Ledoux [12] and Bobkov and Zegarlinski [13]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov and Milman [22].

In the paper we study closures of classes of log–concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class K. In particular we prove that if one starts from one–dimensional log–concave measures, one obtains no non– trivial uniform mesures on convex bodies.

In this paper we give an optimal O(n log n) time and O(n) space algorithm to compute the rectilinear convex layers of a set S of n points on the plane. We also compute the rotation of S that minimizes the number of rectilinear convex layers in O(n log n) time and O(n) space.

In this manuscript, we introduce concepts of (m1,m2)-logarithmically convex (AG-convex) functions and establish some Hermite-Hadamard type inequalities of these classes of functions.

Let K, D be convex centrally symmetric bodies in R. Let k < n and let dk(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define ∆(k, n) = sup dk(K, D), where the supremum is taken over all n−dimensional convex symmetric bodies K, D. We prove that for any k < n ∆(k, n) ∼log n {√ k if k ≤ n k2 n if k > n , where A ∼log n B means that 1/(C log n) ·A ≤ B ≤ (C ...

We introduce a new method of proving lower bounds on the depth of algebraic d-degree decision (resp. computation) trees and apply it to prove a lower bound ~2 (log N) (resp. f2 (log N/log log N)) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N > ( n d ) ~( ' ) (resp. N > nU<n)). This bound apparently does not follow from the methods...