Conditional Convexity

DEA with efficiency classification preserving conditional convexity

Near convexity, metric convexity, and convexity

It is shown that a subset of a uniformly convex normed space is nearly convex if and only if its closure is convex. Also, a normed space satisfying a mild completeness property is strictly convex if and only if every metrically convex subset is convex. 1 Classical and constructive mathematics The arguments in this paper conform to constructive mathematics in the sense of Errett Bishop. This mea...

Projective Convexity in P Implies Grassmann Convexity

In this note we introduce the notion of Grassmann convexity analogous to the wellknown notion of convexity for curves in real projective spaces. We show that the curve in G2,4 osculating to a convex closed curve in P is Grassmann-convex. This proves that the tangent developable (i.e. the hypersurface formed by all tangents) of any convex curve in P has the ‘degree’ equal to 4. Here by ‘degree’ ...

Perceptual Convexity

Often objects which are not convex in the mathematical sense are treated as`perceptually convex'. We present an algorithm for recognition of the perceptual convexity of a 2-d contour. We start by reducing the notion ofà contour is perceptually convex' to the notion ofà contour is Y-convex'. The latter reeects an absence of large concavities in the OY direction of an XOY frame. Then we represent...

Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...