### 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, || · ||) is an inner product space; (2) f : X → ℝ is strongly convex with modulus c > 0 if and only if f - c|| · ||^{2} is convex; (3) || · ||^{2} is strongly convex with modulus 1.

Original language | English |
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Pages (from-to) | 83-87 |

Number of pages | 5 |

Journal | Banach Journal of Mathematical Analysis |

Volume | 5 |

Issue number | 1 |

Publication status | Published - 2011 |

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### Keywords

- Inner product space
- Strongly convex function
- Strongly midconvex function

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis

### Cite this

*Banach Journal of Mathematical Analysis*,

*5*(1), 83-87.

**Characterizations of inner product spaces by strongly convex functions.** / Nikodem, Kazimierz; Páles, Z.

Research output: Contribution to journal › Article

*Banach Journal of Mathematical Analysis*, vol. 5, no. 1, pp. 83-87.

}

TY - JOUR

T1 - Characterizations of inner product spaces by strongly convex functions

AU - Nikodem, Kazimierz

AU - Páles, Z.

PY - 2011

Y1 - 2011

N2 - 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, || · ||) is an inner product space; (2) f : X → ℝ is strongly convex with modulus c > 0 if and only if f - c|| · ||2 is convex; (3) || · ||2 is strongly convex with modulus 1.

AB - 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, || · ||) is an inner product space; (2) f : X → ℝ is strongly convex with modulus c > 0 if and only if f - c|| · ||2 is convex; (3) || · ||2 is strongly convex with modulus 1.

KW - Inner product space

KW - Strongly convex function

KW - Strongly midconvex function

UR - http://www.scopus.com/inward/record.url?scp=78649794965&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649794965&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:78649794965

VL - 5

SP - 83

EP - 87

JO - Banach Journal of Mathematical Analysis

JF - Banach Journal of Mathematical Analysis

SN - 1735-8787

IS - 1

ER -