Differentiability of the Farthest-Point Distance Function

S. Galperin (Russia)


farthest-point distance function, representing measure, Chebyshev center, pattern recognition.


One of the common problems in pattern recognition is a problem of selection of the best representative of a set. For a subset E of n-dimensional euclidean space n R , one of interesting points, representing E, is a Chebyshev center, a point n Ex R∈ such that sup | | inf sup | |nE x Ry E y E x y x y ∈∈ ∈ − = − . In other words, Ex brings infimum to the function ( ) sup | |E y E d x x y ∈ = − , which is called a farthest-point distance function. From the viewpoint of finding the Chebyshev center, differentiability of ( )Ed x seems to be of interest. In this paper, we give an elementary proof of the criterion of differentiability of ( )Ed x on the open set. We also prove reasonable sufficient condition of k-differentiability of ( )Ed x for sets with boundary smooth enough on the plane, and get some consequences of these results.

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