## Banach Journal of Mathematical Analysis

### Characterizations of inner product spaces by strongly convex functions

#### 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 $1$.

#### Article information

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

Dates
First available in Project Euclid: 14 August 2011

https://projecteuclid.org/euclid.bjma/1313362982

Digital Object Identifier
doi:10.15352/bjma/1313362982

Mathematical Reviews number (MathSciNet)
MR2738522

Zentralblatt MATH identifier
1215.46016

#### Citation

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. https://projecteuclid.org/euclid.bjma/1313362982

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