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

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.

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