Knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering  in which there is no need to  be peculiar about how to select initial values. Due to properly converting the task of optimization to an equivalent...

T his article presents a quantitative and objective approach to cat ganglion cell characterization and classification. The combination of several biologically relevant features such as diameter, eccentricity, fractal dimension, influence histogram, influence area, convex hull area, and convex hull diameter are derived from geometrical transforms and then processed by three different clustering ...

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the Lp-affine surface area for convex bodies. Here, we introduce a functional form of this concept, for log concave and s-concave functions. We show that the new functional form is a generalization of the original Lp-affi...

The total diameter of a closed planar curve C ⊂ R is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when C is a circle. We also generalize these results to m dimension...

We consider the problem of minimizing a functional (like the area, perimeter, surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér-Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of cons...

We proved that a finite commuting Boyd-Wong type contractive family with equicontinuous words have the approximate common fixed point property. We also proved that given X Ă R, compact and convex subset, F : X Ñ X a compact-and-convex valued Lipschitz correspondence and g an isometry on X, then gF “ F g implies F admits a Lipschitz selection commuting with g.

Given a convex polygon in the plane, we are interested in triangulations of its interior, i.e. maximal sets of nonintersecting diagonals that subdivide the interior of the polygon into triangles. The MaxMin area triangulation is the triangulation of the polygon that maximizes the area of the smallest area triangle in the triangulation. There exists a dynamic programming algorithm that computes ...

Many applications in the area of production and statistical estimation are problems of convex optimization subject to ranking constraints that represent a given partial order. This problem – which we call the convex cost closure problem, or (CCC) – is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem. For a (CCC) problem on n variables and m ...

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

Let us introduce the notation used throughout the paper. For any notions related to convexity in this paper, consult R. Schneider [8]. We write o to denote the origin in the Euclidean space En, and ‖ · ‖ to denote the corresponding Euclidean norm. Given a set X ⊂ En, the affine hull and the convex hull of X are denoted by affX and convX , respectively, moreover the interior of X is denoted by i...