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

Abstract:

A set S in ${R^d}$ is said to be m-convex, $m \geqslant 2$, 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 ${R^d}$, and assume that the set of lnc points of S lies in a hyperplane. Then S is a union of $f(m)$ or fewer convex sets, where f is defined inductively as follows: $f(2) = 1,f(3) = 2$, and $f(m) = f(m - 2) + 3$ for $m \geqslant 4$. Moreover, for $d \geqslant 3$, an example reveals that the best bound is no lower than $g(m)$, where $g(m) = f(m)$ for $2 \leqslant m \leqslant 5$ and for $m = 7$, and $g(m) = g(m - 3) + 4$ otherwise.