Some remarks on the space of differences of sublinear functions

Volume 22 / 1994

Sven Bartels, Diethard Pallaschke Applicationes Mathematicae 22 (1994), 419-426 DOI: 10.4064/am-22-3-419-426

Abstract

Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X,‖·‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = ℝ^n this construction yields a norm such that D(ℝ^n) becomes a Banach space. Furthermore, we show that for a real Banach space with a smooth dual every sublinear Lipschitzian function can be expressed by the Fenchel conjugate of the farthest point mapping to its subdifferential at the origin. This leads to a simple family of sublinear functions which contains an exhaustive family of upper convex approximations for any quasidifferentiable function.

Authors

  • Sven Bartels
  • Diethard Pallaschke

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