The purpose of this paper is to characterize the Banach spaces and the locally convex spaces E for which bounded additive measures or bounded σadditive measures with values in L(E, F ), the space of continuous linear maps from E into F , are of bounded semi-variation for any Banach space or locally convex space F . This paper gives an answer to a problem posed by D.H. Tucker in [6].

The convex hull of any subset o f vertices of an n-dimensional hypercube contains no other vertex of the hypercube. This result permits the application of some theorems of n-dimensional geometry lo digital reed-forward neural networks. Also. the construction Of the convex hull is proposed as an alternative to more traditional learning algorithms. Some preliminary simulation results are reponed.

We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O (n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.

We consider the problem of maximizing the Lebesgue measure of the convex hull of a connected compact set of prescribed onedimensional Hausdorff measure. In dimension two, we prove that the only solutions are semicircles. In higher dimension, we prove some isoperimetric inequalities for the convex hull of connected sets, we focus on a classical open problem and we discuss a new possible approach.

Often the structure of discrete sets can be described in terms of a closure operator. When each closed set has a unique minimal generating set (as in convex geometries in which the extreme points of a convex set generate the closed set), we have an antimatroid closure space. In this paper, we show there exist antimatroid closure spaces of any size, of which convex geometries are only a sub-fami...

Of late, researchers appear to be intrigued with the question; Given a set of points, what is the region occupied by them? The answer appears to be neither straight forward nor unique. Convex hull, which gives a convex enclosure of the given set, concave hull, which generates nonconvex polygons and other variants such as α-hull, poly hull, r-shape and s-shape etc. have been proposed. In this pa...

In this note we prove that if K is a compact set of m×n matrices containing an isolated point X with no rank-one connection into the convex hull of K \ {X}, then the rank-one convex hull separates as K = ( K \ {X} )rc ∪ {X}. This is an extension of a result of P. Pedregal, which holds for 2× 2 matrices.

We present a numerically stable and time and space complexity optimal algorithm for constructing a convex hull for a set of points on a plane. In contrast to already existing numerically stable algorithms which return only an approximate hull, our algorithm constructs a polygon that is truly convex. The algorithm is simple and easy to implement.

Abstract. Consider n points X1, . . . ,Xn in R and denote their convex hull by Π. We prove a number of inclusion-exclusion identities for the system of convex hulls ΠI := conv(Xi : i ∈ I), where I ranges over all subsets of {1, . . . , n}. For instance, denoting by ck(X) the number of k-element subcollections of (X1, . . . ,Xn) whose convex hull contains a point X ∈ R, we prove that c1(X) − c2(...