Abstract
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 $1$.
Citation
Kazimierz Nikodem. Zsolt Pales. "Characterizations of inner product spaces by strongly convex functions." Banach J. Math. Anal. 5 (1) 83 - 87, 2011. https://doi.org/10.15352/bjma/1313362982
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