-- Antoine de Saint-Exupery
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Abstracts
Spring Conference on Analysis, May - June 2005
Pavel Shvartsman, Technion, Haifa:
The Whitney Extension Problem and Helly's intersection theorem
In this series of talks we will discuss the following problem posed by H. Whitney in 1934:Let f be an arbitrary function defined on a subset S of the multidimensional space and let k be a positive integer. What is a necessary and sufficient condition for f to be the restriction to S of a Ck-function?
The main result states that at least for k = 1,2 the trace space has so-called "finiteness" property, i.e., the restriction of a function f to all of S can be completely determined by restrictions of f to every finite subset of S consisting of a prescribed number of points. Recently C. Fefferman has proved that the finiteness property is true for k>2.
We will demonstrate surprising connections of this property with different problems of Convex Geometry (e.g., Helly's intersection theorem, Lipschitz selections of set-valued functions, the Steiner point of a convex body etc.) and Riemannian Geometry (e.g., Lipschitz functions on hyperbolic spaces). We will also present an application of Helly-type results to the K-divisibility problem of the real interpolation method and to the problem of extension of Lipschitz mappings.