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Zbl 1215.46016
Nikodem, Kazimierz; Páles, Zsolt
Characterizations of inner product spaces by strongly convex functions. (English)
[J] Banach J. Math. Anal. 5, No. 1, 83-87, electronic only (2011). ISSN 1735-8787/e

Summary: New characterizations of inner product spaces among normed spaces involving the notion of strong convexity are given. In particular, it is shown that the following conditions are equivalent: (1) $(X,\|\cdot\|)$ is an inner product space; (2) $f : X\to\Bbb R$ is strongly convex with modulus $c>0$ if and only if $f-c\|\cdot\|^2$ is convex; (3) $\|\cdot\|^2$ is strongly convex with modulus 1.

MSC 2000:: *46C15 Characterizations of Hilbert spaces

Keywords: inner product space; strongly convex function; strongly midconvex function

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Zentralblatt MATH Berlin [Germany]