An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by ...

We study the existence of Stackelberg equilibrium points on strategy sets which are geodesic convex in certain Riemannian manifolds by using metric projection arguments. The present results extend those obtained in Nagy [J. Global Optimization (2013)] in the Euclidean context.

We study the quantum ( $$C^*$$ ) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, -extreme POVMs countable spaces (in particular for finite sets) are always spectral (normalized projection measures). More generally shown atomic spectral. A Krein–Milman type theorem ha...

We define the notions of strong convexity and strong visibility. These notions generalize standard convexity and visibility, as well as several types of nontraditional convexity, such as iso-oriented rectangles and C-oriented polygons. We explore the properties of strong convexity and strong visibility in two and three dimensions. In particular, we establish analogs of the following properties ...

Let f ∈ C[−1, 1] change its convexity finitely many times in the interval, say s times, at Ys : −1 < y1 < · · · < ys < 1. We estimate the degree of approximation of f by polynomials of degree n, which change convexity exactly at the points Ys. We show that provided n is sufficiently large, depending on the location of the points Ys, the rate of approximation is estimated by the third Ditzian–To...

A classical result of Alexsandrov [1] asserts that convex functions in R are twice differentiable a.e., (see also [3], [8] for more modern treatments). It is well known that Sobolev functions u ∈ W , for p > n/2 are twice differentiable a.e.. The following weaker notion of convexity known as k-convexity was introduced by Trudinger and Wang [12, 13]. Let Ω ⊂ R be an open set and C(Ω) be the clas...

A new five-point binary subdivision scheme with high continuity and convexity preservation is proposed in this paper. It is shown that the limit curves are C (k = 0, 1, . . . , 7) continuous for the certain range of the parameter. The range of the parameter for the property of convexity preservation of the limit curves is also provided. Experimental results demonstrate the efficiency and flexib...