J. Aust. Math. Soc.  82 (2007), 111-121
Semicontinuous functions and convex sets in C (K ) spaces

J. P. Moreno
  Dpto. Matemáticas
  Facultad de Ciencias
  Universidad Autónoma de Madrid
  Madrid 28049

The stability properties of the family $\mathcal M$ of all intersections of closed balls are investigated in spaces C (K ), where K is an arbitrary Hausdorff compact space. We prove that $\mathcal M$ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that $\mathcal M$ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C [0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.
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