Open Access

On moduli of convexity in Banach spaces

Journal of Inequalities and Applications20052005:695306

DOI: 10.1155/JIA.2005.423

Received: 15 September 2003

Published: 31 August 2005

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq1_HTML.gif be a normed linear space, https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq2_HTML.gif an element of norm one, and https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq4_HTML.gif the local modulus of convexity of https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq5_HTML.gif . We denote by https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq6_HTML.gif the greatest https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq7_HTML.gif such that for each closed linear subspace https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq8_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq9_HTML.gif the quotient mapping https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq10_HTML.gif maps the open https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq11_HTML.gif -neighbourhood of https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq12_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq13_HTML.gif onto a set containing the open https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq14_HTML.gif -neighbourhood of https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq15_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq16_HTML.gif . It is known that https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq17_HTML.gif . We prove that there is no universal constant https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq18_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq19_HTML.gif , however, such a constant https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq20_HTML.gif exists within the class of Hilbert spaces https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq21_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq22_HTML.gif is a Hilbert space with https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq23_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2FJIA.2005.423/MediaObjects/13660_2003_Article_1543_IEq24_HTML.gif .

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia

Copyright

© Reif. 2005