Geodesic
(D-2)-Extreme Points and a Helly-Type Theorem for Starshaped Sets
Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 703-713

Voir la notice de l'article provenant de la source Cambridge University Press

We begin with some preliminary definitions. Let S be a subset of R d. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by S x = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{S x: x in S}.Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in R d, d ≧ 2, then ker S = ⋂{S e: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected. 
Breen, Marilyn. (D-2)-Extreme Points and a Helly-Type Theorem for Starshaped Sets. Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 703-713. doi: 10.4153/CJM-1980-055-2
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