A bound for decompositions of 𝑚-convex sets whose LNC points lie in a hyperplane

Abstract

A set S in R d {R^d} is said to be m-convex, m ⩾ 2 m \geqslant 2 , if and only if for every m points in S, at least one of the line segments determined by these points lies in S. Let S denote a closed m-convex set in R d {R^d} , and assume that the set of lnc points of S lies in a hyperplane. Then S is a union of f ( m ) f(m) or fewer convex sets, where f is defined inductively as follows: f ( 2 ) = 1 , f ( 3 ) = 2 f(2) = 1,f(3) = 2 , and f ( m ) = f ( m − 2 ) + 3 f(m) = f(m - 2) + 3 for m ⩾ 4 m \geqslant 4 . Moreover, for d ⩾ 3 d \geqslant 3 , an example reveals that the best bound is no lower than g ( m ) g(m) , where g ( m ) = f ( m ) g(m) = f(m) for 2 ⩽ m ⩽ 5 2 \leqslant m \leqslant 5 and for m = 7 m = 7 , and g ( m ) = g ( m − 3 ) + 4 g(m) = g(m - 3) + 4 otherwise.