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 wellknown 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. Repovs.
Moduli of uniform convexity for convex sets / C.A. De Bernardi, L. Vesely. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - (2024), pp. 1-10. [Epub ahead of print] [10.1007/s00013-024-02031-8]
Moduli of uniform convexity for convex sets
C.A. De BernardiPrimo
;L. Vesely
Ultimo
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 wellknown 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. Repovs.
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