A note on self-extremal sets in \(L_p(\Omega)\) spaces (Q2368476)

A nonempty bounded set in a Banach space is said to be self-extremal if its relative Chebyshev radius with respect to the closed convex hull coincides with the diameter times the self-Jung constant (which is defined as the supremum of all relative Chebyshev radii of sets of diameter one). Let \((\Omega, \mu)\) be a \(\sigma\)-finite measure space. It is known that in the infinite-dimensional situation, for \(1<p< \infty\) the self-Jung constant of \(L_p(\Omega)\) is \( \max \{2^{\frac{1}{p-1}}, 2^{\frac {-1}{p}}\}\). The main result of this paper is that for self-extremal sets in these spaces, the Kuratowski measure of noncompactness coincides with the diameter.