The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry

Abstract

We provide a precise characterization of the weak sequential closure of nonempty, closed, decomposable sets in Lebesgue spaces. Therefore, we have to distinguish between the purely atomic and the nonatomic regime. In the latter case, we get a convexification effect which is related to Lyapunov’s convexity theorem, and in the former case, the weak sequential closure equals the strong closure. The characterization of the weak sequential closure is utilized to compute the limiting normal cone to nonempty, closed, decomposable sets in Lebesgue spaces. Finally, we give an example for the possible nonclosedness of the limiting normal cone in this setting.

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Correspondence to Gerd Wachsmuth.

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Mehlitz, P., Wachsmuth, G. The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry. Set-Valued Var. Anal 27, 265–294 (2019). https://doi.org/10.1007/s11228-017-0464-1

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Keywords

  • Decomposable set
  • Lebesgue spaces
  • Limiting normal cone
  • Measurability
  • Weak sequential closure

Mathematics Subject Classification (2010)

  • 49J53
  • 28B05
  • 90C30