Generalized Centers of Finite Sets in Banach
نویسنده
چکیده
We study mainly the class (GC) of all real Banach spaces X such that the set E f (a) of the minimizers of the function X 3 x 7 ! f(kx ? a 1 k;: : : ; kx ? a N k) is nonempty whenever N is a positive integer, a 2 X N , and f is a continuous monotone coercive function on 0;+1 N. For particular choices of f, the set E f (a) coincides with the set of Chebyshev centers of the set fa i : i = 1;: :: ; Ng or with the set of its medians. The class (GC) is stable under making c 0-, ` p-and similar sums. Under some geometric conditions on X, the function spaces C b (T;X) or L p (; X) belong to (GC). One of the main tools is a theorem which asserts that, in the deenition of the class (GC), one can restrict himself to the functions f of the type f(1
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