Let S be a finite set with n elements in a real linear space. Let JS be a set of n intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved that each homothet of conv S that is contained in aff S can be obtained using this operator. A vari...

In this paper, we study properties of general closed convex sets that determine the closed-ness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special class of convex sets such as pointed cones, stric...

In this paper we characterise the maximal convex subsets of the (non-convex) rate region in 802.11 WLANs. In addition to being of intrinsic interest as a fundamental property of 802.11 WLANs, this characterisation can be exploited to allow the wealth of convex optimisation approaches to be applied to 802.11 WLANs.

We consider the following basic geometric problem: Given ∈ (0, 1/2), a 2-dimensional figure that consists of a black object and a white background is -far from convex if it differs in at least an fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is -far from convex are needed to detect a violation of convexity wi...

Consider a convex domain B of R 3. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D ′ are convex domains with D bounded and the closure of D contained in D ′ then any minimal disk whose boundary lies in the boundary of D, can be approximated in any compact subdomain of D by a complete minimal disk which is proper in D ′. W...

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the ...

The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f : X → R defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E, we draw the possibility to state our results for arbitrary real linear spa...

Let ∆ be a thick dual polar space and F a convex subspace of diameter at least 2 of ∆. Every hyperplane G of the subgeometry F̃ of ∆ induced on F will give rise to a hyperplane H of ∆, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of ∆ and if eF is the full embedding of F̃ ...

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem inf { ess sup x∈Ω f(∇u(x)) : u ∈ u0 +W 1,∞ 0 (Ω) } , when the supremand f is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Severa...

We consider the class of semi-stable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n− 2. When the domain is convex, we establish a priori Lp and H1 0 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead...