On Poidge-Convexity

Let

be a family of sets in

\mathbb{R}^d\ (\mathrm{always}\ d\geq 2) R d   ( a l w a y s   d ≥ 2 )

. A set

M\subset\mathbb{R}^d M ⊂ R d

is called

\mathcal{F} F

-convex, if for any pair of distinct points

x, y \in M x , y ∈ M

, there is a set

F\in \mathcal{F} F ∈ F

such that

x, y \in F x , y ∈ F

and

F \subset M F ⊂ M

. We obtain the poidge-convexity, when

\mathcal{F} F

consists of all unions

\{x\}\cup \sigma { x } ∪ σ

, called poidges, where

x x

is a point,

\sigma σ

a line-segment, and

\mathrm{ conv}(\{x\}\cup \sigma) c o n v ( { x } ∪ σ )

a right triangle. In this paper we first present several new results on the poidge-convexity of various sets, such as unions of line-segments, fans, cones and cylinders, complements of some given sets and not simply connected sets. Then, we investigate the poidge-convex completion of compact convex sets, trying to determine the minimal number of points necessary to be added to make them poidge-convex.