Let $C$ be a proper, closed subset with nonempty interior in a normed space $X$. We define four variants of modulus of convexity for $C$ and prove that they all coincide. This result, which is classical and well-known for $C=B_X$ (the unit ball of $X$), requires a less easy proof than the particular case of $B_X$. We also show that if the modulus of convexity of $C$ is not identically null then $C$ is bounded. This extends a result by M.V.~Balashov and D.~Repov\v{s}.
De Bernardi, C. A., Veselý, L., Moduli of uniform convexity for convex sets, <<ARCHIV DER MATHEMATIK>>, 2024; 123 (4): 413-422. [doi:10.1007/s00013-024-02031-8] [https://hdl.handle.net/10807/297287]
Moduli of uniform convexity for convex sets
De Bernardi, Carlo Alberto;
2024
Abstract
Let $C$ be a proper, closed subset with nonempty interior in a normed space $X$. We define four variants of modulus of convexity for $C$ and prove that they all coincide. This result, which is classical and well-known for $C=B_X$ (the unit ball of $X$), requires a less easy proof than the particular case of $B_X$. We also show that if the modulus of convexity of $C$ is not identically null then $C$ is bounded. This extends a result by M.V.~Balashov and D.~Repov\v{s}.
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