Abstract
AbstractLet 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 $$C=B_X$$
C
=
B
X
(the unit ball of X), requires a less easy proof than the particular case of $$B_X.$$
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. Repovš.
Funder
Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
Università degli Studi di Milano
Agencia Estatal de Investigación
Publisher
Springer Science and Business Media LLC
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