On starshapedness in products of interval spaces

Later a bouquet of Krasnosel 'skii-type theorems was discovered; see [1,2, 3, 4,16,17]. Apart from the usual line segment visibility and starshapedness several other notions of starshaped set and visibility have been investigated in the past years: d-starshaped sets in metric and normed spaces [14,15,16,17], starshaped sets in graphs [14,171, staircase visibility [5,10,12], rectangular visibility [10,11,14] etc. More recently, starshaped sets have been studied from the viewpoint of abstract convexity [7, 8,15,16]. In many of the investigations of axiomatic convexity theory the numbers of Carath6odory, Helly and Radon of products of convexity spaces have been studied; see [13,16,18]. Motivated by these results, this note is devoted to Krasnosel 'skii 's number of product of interval spaces. The following notations are taken from [16,18]. A family cg of subsets of a set X is called a convexity on X if 6 and X are in ~g and cg is closed under arbitrary intersections. The pair (X, cr is a convex structure, and members of ~g are called convex sets. Each A ~ X is contained in a smallest convex set, the convex hull of A or cony IA) for short. The convex hull of two points is called a segment. An operator I : X 2 ~ 2 x is called an interval operator if it has the following properties for all a, b e X: