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...

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 ...

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...

Let Ω be the upper half plane {(x, y) ∈ R, x > 0, y ∈ R}. Define the Laplacian on Ω to be ∆D = ∂ 2 x + (1 + x)∂ 2 y , together with Dirichlet boundary conditions on ∂Ω: one may easily see that Ω, with the metric inherited from ∆D, is a strictly convex domain. We shall prove that, in such a domain Ω, Strichartz estimates for the wave equation suffer losses when compared to the usual case Ω = R, ...

A restricted-orientation convex set, also called an O-convex set, is a set of points whose intersection with lines from some fixed set is empty or connected. The notion of O-convexity generalizes standard convexity and orthogonal convexity. We explore some of the basic properties of O-convex sets in two and higher dimensions. We also study O-connected sets, which are a subclass of O-convex sets...

Let Ω ⊆ Rn be a strictly convex domain and let φ ∈ C2(Ω) be a convex function such that λ ≤ detD2φ ≤ Λ in Ω. The linearized Monge– Ampère equation is LΦu = trace(ΦD u) = f, where Φ = (detD2φ)(D2φ)−1 is the matrix of cofactors of D2φ. We prove that there exist p > 0 and C > 0 depending only on n, λ,Λ, and dist(Ω′,Ω) such that ‖Du‖Lp(Ω′) ≤ C(‖u‖L∞(Ω) + ‖f‖Ln(Ω)) for all solutions u ∈ C2(Ω) to the...

a compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. directed hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. sets of extrema of birkhoff polyhedra are mapped to tours ...

‎In this paper, generalized convex contractions on orthogonal metric spaces are stablished in whath  might be called their  definitive versions. Also, we show that there are examples which show that our main theorems are  genuine generalizations of Theorem 3.1 and 3.2 of [M.A. Miandaragh, M. Postolache and S. Rezapour,  {it Approximate fixed points of generalized convex contractions}, Fixed Poi...