We show that the abstract Voronoi diagram of n sites in the plane can be constructed in time O(n log n) by a randomized algorithm. This yields an alternative, but simpler, O(n log n) algorithm in many previously considered cases and the first O(n log n) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of b...

Given a set of n points P in the plane, the first layer L1 of P is formed by the points that appear on P ’s convex hull. In general, a point belongs to layer Li, if it lies on the convex hull of the set P \ ⋃ j<i{Lj}. The convex layers problem is to compute the convex layers Li. Existing algorithms for this problem either do not achieve the optimal O (n log n) runtime and linear space, or are o...

Let μ = ρdx be a Borel measure on Rd. A Borel set A ⊂ R is a solution of the isoperimetric problem if for any B ⊂ R satisfying μ(A) = μ(B) one has μ(∂A) ≤ μ(∂B), where μ(∂A) = ∫ ∂A ρ dHd−1 is the corresponding surface measure. There exists only a small number of examples where the isoperimetric problem has an exact solution. The most important case is given by Lebesgue measure λ on R, the solut...

We give novel algorithms for stochastic strongly-convex optimization in the gradient oracle model which return a O( 1 T )-approximate solution after T iterations. The first algorithm is deterministic, and achieves this rate via gradient updates and historical averaging. The second algorithm is randomized, and is based on pure gradient steps with a random step size. This rate of convergence is o...

It was open for more than eight years to obtain an algorithm for computing the convex hull of a set of n sorted points in sub-logarithmic time on a reconfigurable mesh of size p n pn. Our main contribution is to provide the first breakthrough: we propose an almost optimal algorithm running in O((log logn)2) time on a reconfigurable mesh of size p n pn. With slight modifications this algorithm c...

‎In this paper‎, ‎we generalize the proximal point algorithm to complete CAT(0) spaces and show‎ ‎that the sequence generated by the proximal point algorithm‎ $w$-converges to a zero of the maximal‎ ‎monotone operator‎. ‎Also‎, ‎we prove that if $f‎: ‎Xrightarrow‎ ‎]-infty‎, +‎infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal...

We derive a lower bound on the differential entropy of a log-concave random variable X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure |x− x̂|,...

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

We study competitive economy equilibrium computation. We show that, for the first time, the equilibrium sets of the following two markets: 1. A mixed Fisher and ArrowDebreu market with homogeneous and log-concave utility functions; 2. The Fisher and Arrow-Debreu markets with several classes of concave non-homogeneous utility functions; are convex or log-convex. Furthermore, an equilibrium can b...