Abstract
We give a geometrical interpretation of the Brudnyi-KrugljakK-divisibility theorem—one of the fundamental results of modern interpolation theory of Banach spaces. We show that this result is closely connected with a curious intersection theorem which can be formulated in the spirit of Helly’s classical theorem. LetB 0,B 1 be two closed convex balanced subset of a Banach spaceX. We prove that under a wide range of various conditions the family of setsB = {B =sB 0 +tB 1 +c;s, t ∈R,c ∈X} possesses the following intersection property:
LetB′ be a subfamily ofB such that every two sets fromB′ have a common point. Then ∩ B∈B′ γ oB ≠ 0, where γ>0 is an absolute constant (γ ≤ 7 + 4 √2) and the symbol γ oB denotes a dilation ofB with respect to its center by a factor of γ.
As a consequence we obtain a generalization of theK-divisibility theorem for sums of two elements.
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Supported by the Center for Absorption in Science, Israel Ministry of Immigrant Absorption and by grant No. 95-00225 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.
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Shvartsman, P. A geometrical approach to theK-divisibility problem. Isr. J. Math. 103, 289–318 (1998). https://doi.org/10.1007/BF02762277
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DOI: https://doi.org/10.1007/BF02762277