In this paper we prove the following results concerning the product f1f2 of two concave functions f1and f2 defined in a non empty, compact, convex set K of R. The first result states that necessary and sufficient condition for the product f1f2 of two linear affine functions to be concave is that they are Gonzi. Finally the main result says that a necessary and sufficient condition for the produ...

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincaré inequalities for such functions. This leads naturally to the concept of f -divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valua...

Bounding hull, such as convex hull, concave hull, alpha shapes etc. has vast applications in different areas especially in computational geometry. Alpha shape and concave hull are generalizations of convex hull. Unlike the convex hull, they construct non-convex enclosure on a set of points. In this paper, we introduce another generalization of convex hull, named alpha-concave hull, and compare ...

In this paper, we study the approximation and estimation of s-concave densities via Rényi divergence. We first show that the approximation of a probability measure Q by an s-concave density exists and is unique via the procedure of minimizing a divergence functional proposed by [Ann. Statist.38 (2010) 2998-3027] if and only if Q admits full-dimensional support and a first moment. We also show c...

In this paper we propose a regularized relaxation based graph matching algorithm. The graph matching problem is formulated as a constrained convex quadratic program, by relaxing the permutation matrix to a doubly stochastic one. To gradually push the doubly stochastic matrix back to a permutation one, a simple weighted concave regular term is added to the convex objective function. The concave ...

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the Lp-affine surface area for convex bodies. Here, we introduce a functional form of this concept, for log concave and s-concave functions. We show that the new functional form is a generalization of the original Lp-affi...

R d is log-concave if p = e where φ :Rd → [−∞,∞) is concave. We denote the class of all such densities p on R by Pd,0. Log-concave densities are always unimodal and have convex level sets. Furthermore, log-concavity is preserved under marginalization and convolution. Thus, the classes of log-concave densities can be viewed as natural nonparametric extensions of the class of Gaussian densities. ...

This paper proposes a method based on linear programming techniques to treat quasi-concave and non-concave fuzzy multi-objective programming (FMOP) problems. The proposed method initially presents a piecewise linear expression to interpreting a quasi-concave membership function. Then we 7nd the convex-type break points and transform all quasiconcave membership functions into concave functions. ...

A function F defined on all subsets of a finite ground set E is quasiconcave if F (X∪Y ) ≥ min{F (X), F (Y )} for all X, Y ⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concav...