Banach Journal of Mathematical Analysis

Characterizations of inner product spaces by strongly convex functions

Kazimierz Nikodem and Zsolt Pales

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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) $f:X\to \R$ 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 $1$.

Article information

Banach J. Math. Anal. Volume 5, Number 1 (2011), 83-87.

First available in Project Euclid: 14 August 2011

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Zentralblatt MATH identifier

Primary: 46C15: Characterizations of Hilbert spaces
Secondary: 26B25: Convexity, generalizations 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx]

inner product space strongly convex function strongly midconvex function


Nikodem, Kazimierz; Pales, Zsolt. Characterizations of inner product spaces by strongly convex functions. Banach J. Math. Anal. 5 (2011), no. 1, 83--87. doi:10.15352/bjma/1313362982.

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