Borel classes of sets of extreme and exposed points in ℝⁿ

Holický, Petr; Keleti, Tamás

doi:10.1090/S0002-9939-05-07743-9

Borel classes of sets of extreme and exposed points in $\mathbb {R}^n$
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by Petr Holický and Tamás Keleti PDF

Proc. Amer. Math. Soc. 133 (2005), 1851-1859 Request permission

Abstract:

It is known that the sets of extreme and exposed points of a convex Borel subset of $\mathbb {R}^n$ are Borel. We show that for $n\ge 4$ there exist convex $G_{\delta }$ subsets of $\mathbb {R}^n$ such that the sets of their extreme and exposed points coincide and are of arbitrarily high Borel class. On the other hand, we show that the sets of extreme and of exposed points of a convex set $C\subset \mathbb {R}^3$ of additive Borel class $\alpha$ are of ambiguous Borel class $\alpha +1$. For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of $C\cap \partial C$ are of the additive Borel class $\alpha$ if $C\subset \mathbb {R}^3$ is a convex set of additive Borel class $\alpha$.

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