Let X be a real normed space, A be a non-singleton compact subset of X, m be a probability measure on the Borel -algebra of A such that for each open set V in X with V \ A 6= ?, m(V \A) > 0. For each x, y 2 X, let A(y, x) = {a 2 A: ky −ak < kx−ak}, WA(y, x) = m(A(y, x)),WA(x) = supz2X WA(z, x). We prove that if X is finite-dimensional and x is an interior point of A, thenWA(x) < 1.
Simpson points in normed spaces.
CASINI, EMANUELE GIUSEPPE
1996-01-01
Abstract
Let X be a real normed space, A be a non-singleton compact subset of X, m be a probability measure on the Borel -algebra of A such that for each open set V in X with V \ A 6= ?, m(V \A) > 0. For each x, y 2 X, let A(y, x) = {a 2 A: ky −ak < kx−ak}, WA(y, x) = m(A(y, x)),WA(x) = supz2X WA(z, x). We prove that if X is finite-dimensional and x is an interior point of A, thenWA(x) < 1.
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