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. LetB0,B1 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 =sB0 +tB1 +c;s, t ∈R,c ∈X} possesses the following intersection property:
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