**
J. Rákosník (ed.), **

Function spaces, differential operators and nonlinear analysis.

Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.

Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996

p. 195 - 200

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Normal structure and modulus of $\boldkey U$-convexity in Banach spaces

##
Ji Gao

Department of Mathematics, Community College of Philadelphia, 1700 Spring Garden Street, Philadelphia, PA 19130-3991, U.S.A. ccp_math@shrsys.hslc.org

**Abstract:**
Let $X$ be a normed linear space, and let $S(X)$ be the unit sphere
of $X$. A geometric parameter of $X$, $J(X)$, is introduced; the
properties of $J(X)$ and the relationship between $J(X)$ and the
modulus of convexity of $X$ are discussed. The main result is that
either $J(X) < 3/2$ or the value of the modulus of convexity of $X$
at $3/2 > 1/4$ implies uniform normal structure.

An example of the Banach space $X$ with the values of the modulus of
convexity at 1 and 3/2 are 0 and 1/4, respectively, is given. The
unit sphere $S(X)$ is deformed from the hexagon. It shows that the
two conditions which guarantee uniform normal structure are
independent.

**Full text of the article:**

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