A point \( s \) of a closed convex subset \( S \) of \( k \)-space is admissible if there is no \( t\in S \) with \( t_i\leq s_i \) for all \( i=1,\dots \), \( k \), \( t\neq s \). An example is given in which the set \( A \) of admissible points is not closed.
Let \( P \) be the set of vectors \( p=(p_1,\dots \), \( p_k) \) with \( p_i > 0 \) and \( \sum_1^k p_i=1 \), let \( B(p) \) be the set of \( s\in S \) with
\[ (p,s)=\min_{t\in S}(p,t) ,\]
and let \( B=\sum B(p) \), so that \( B \) consists of exactly those points of \( S \) at which there is a supporting hyperplane whose normal has positive components.
\( B\subset A\subset \overline{B} \). If \( S \) is determined by a finite set, there is a finite set \( p_1,\dots \), \( p_N \), with \( p_j\in P \), such that \( B=\sum_{j=1}^N B(p_j) \), so that, since \( B(p) \) is closed for fixed \( p \), \( B = A = \overline{B} \).
