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Journal of Convex Analysis 10 (2003), No. 1, 169--184
Copyright Heldermann Verlag 2003
Convexity and the Natural Best Approximation in Spaces of Integrable Young Measures
Zvi Artstein
Dept. of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel,
zvi.artstein@weizmann.ac.il
Cristian Constantin Popa
Dept. of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel,
popa@wisdom.weizmann.ac.il
The natural best approximation in function spaces singles, out of the
family of best L1-approximation of an integrable function in a
convex set, the element which is the limit as p converges to 1+, of the unique
best Lp-approximation of the function. The present paper extends
the result to convex sets in spaces of integrable Young measures. Such spaces
lack a standard affine structure. In this paper convexity is considered via
a limiting procedure. Consequently, the proof of the existence of a natural
best approximation does not rely on tools like weak convergence, available
in an ordinary function space. Rather, the interplay of compactness and
convexity in the relaxed setting plays a major role.
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