MR2580162 (2011b:46030) Martinón, Antonio A note on measures of nonconvexity. Nonlinear Anal. 72 (2010), no. 6, 3108–3111. (Reviewer: Diana Caponetti), 46B20 (52A05 54B20)

Abstract

Eisenfeld and Lakshmikantham [Yokohama Math. J. 24 (1976), no.1-2, 133-140; MR0425704 (54$\#$13657)] defined the measure of nonconvexity $\alpha(C)$ of a subset $C$ of a Banach space $X$ to be the Hausdorff distance $h(C, {\rm conv} C)$ between the set $C$ and its convex hull. In this note the author, for a nonempty bounded subset $C$ of $X$, defines a measure of nonconvexity $\beta(C)$ as the Hausdorff distance of $C$ to the family $bx(X)$ of all nonempty bounded convex subsets of $X$, i.e. $ \beta(C)= \inf_{K \in bx(X)}h(C,K ). $ The author studies the properties of $\beta$. He shows that $\alpha$ and $ \beta$ are equivalent, but not equal in the general case.