Banach Journal of Mathematical Analysis

Characterizations of inner product spaces by strongly convex functions

Kazimierz Nikodem and Zsolt Pales

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1313362982

Digital Object Identifier
doi:10.15352/bjma/1313362982

Mathematical Reviews number (MathSciNet)
MR2738522

Zentralblatt MATH identifier
1215.46016

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

Keywords
inner product space strongly convex function strongly midconvex function

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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