We extend the recent results of R. Lata la and O. Guédon about equivalence of Lq-norms of logconcave random variables (KahaneKhinchin’s inequality) to the quasi-convex case. We construct examples of quasi-convex bodies Kn ⊂ IRn which demonstrate that this equivalence fails for uniformly distributed vector on Kn (recall that the uniformly distributed vector on a convex body is logconcave). Our e...

hold. This double inequality is known in the literature as the Hermite–Hadamard inequality for convex functions. In recent years many authors established several inequalities connected to this fact. For recent results, refinements, counterparts, generalizations and new Hermite-Hadamard’stype inequalities see [1]–[18]. We recall that the notion of quasi-convex function generalizes the notion of ...

We propose a new approach to the numerical treatment of non(quasi)convex rate-independent evolutionary problems. The main idea is to replace the non(quasi)convex energy density by its polyconvexification. For this problem, first-order optimality conditions are derived and used in finding a discrete solution. The effectiveness of the method is illustrated with some numerical experiments.

In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of nonintegral convex polygons. Define a pseudo-integral polygon, or PIP, to be a convex rational polygon whose Ehrhart quasipolynomial is a polynomial. The numbers of lattice points on the interior and on the boundary of a PIP determin...

It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of...

We show constructively that every quasi-convex uniformly continuous function f : C → R+ has positive infimum, where C is a convex compact subset of Rn. This implies a constructive separation theorem for convex sets.

Suppose τ is a train track on a surface S. Let C (τ) be the set of isotopy classes of simple closed curves carried by τ. Masur and Minsky [2004] prove that C (τ) is quasi-convex inside the curve complex C (S). We prove that the complement, C (S)−C (τ), is quasi-convex.