Sharper approximation of extreme points by far points (Q689713)

When \(K\) is a convex subset of a normed vector space \(E\), a point \(p\) of \(K\) is called a far point if there exists a point \(y\) of \(E\) such that \(\| y-p\|\geq \| y-q\|\) for all \(q\in K\). It was proved by \textit{S. Straszewicz} [Fundam. Math. 24, 139-143 (1935; Zbl 0011.13001)] that if \(E\) is a finite-dimensional Euclidean space and \(K\) is compact, then the set of far points of \(K\) is dense in its set of extreme points. This was extended by the author [Math. Z. 69, 90-104 (1958; Zbl 0079.125)] to an arbitrary strictly convex normed space. Here is the main theorem of the present paper: If \(E\) is an inner-product space, \(K\) is compact, \(p_ 0\) is an extreme point of \(K\), and \(H_ 0\) is a hyperplane supporting \(K\) at \(p_ 0\) such that the intersection \(K\cap H_ 0\) is finite-dimensional, then there is a sequence \(p_ 1,p_ 2,\dots\) of far points of \(K\) converging to \(p_ 0\) and a sequence \(H_ 1,H_ 2,\dots\) of hyperplanes converging to \(H_ 0\) such that for each \(i\), \(H_ i\) supports \(K\) at \(p_ i\). This theorem was motivated by an application to a mathematical model of a problem in reactor design.