# uniformly convex space

A normed space is *uniformly convex* iff $\forall \u03f5>0$ there exists $\delta >0$ that satisfies
for $\parallel x\parallel \le 1$ $\parallel y\parallel \le 1$ and $\parallel x-y\parallel >\u03f5$ $\Rightarrow $ $\parallel \frac{x+y}{2}\parallel \le 1$-$\delta $.

For example it is easily seen that the normed space $({\mathbb{R}}^{2},\parallel .{\parallel}_{2})$ is uniformly convex space.
Also ${L}^{p}$ and ${l}^{p}$ spaces for $$ are uniformly convex, see J.A. Clarkson, ”Uniformly convex spaces”, Trans. Amer. Math. Society, 40 (1936), 396-414.

Title | uniformly convex space |
---|---|

Canonical name | UniformlyConvexSpace |

Date of creation | 2013-03-22 15:13:11 |

Last modified on | 2013-03-22 15:13:11 |

Owner | georgiosl (7242) |

Last modified by | georgiosl (7242) |

Numerical id | 32 |

Author | georgiosl (7242) |

Entry type | Definition |

Classification | msc 46H05 |

Synonym | uniformly convex |