Session 10. Generalized Convexity

Demyanov Difference in infinite dimensional Spaces

Jezry Grzybowski, Adam Mickiewicz University, Poznań, Poland, University of Karlsruhe, Karlsruhe, Germany
This talk is based on a joint work with Ryszard Urbański and Diethard Pallaschke
We generalize the Demyanov difference to the case of real Hausdorff topological vector spaces. For \(A,B \subset X\) we define upper difference \(\mathcal{E}_{A,B}\) as the family \(\mathcal{E}_{A,B} = \{C \in \mathcal{C}(X) | A \subset \overline{B+C}\}\), where \(\mathcal{C}(X)\) is the family of all nonempty closed convex subsets of the topological vector space \(X\). We denote the family of inclusion minimal elements of \(\mathcal{E}_{A,B}\) by \(m\mathcal{E}_{A,B}\). We define a new subtraction by \(A\stackrel{D}{-}B = \overline{\rm conv} \bigcup m\mathcal{E}_{A,B}\). We show that \(A\stackrel{D}{-}B\) is a generalization of Demyanov difference. We prove some clasical properties of the Demyanov difference. For a locally convex vector space \(X\) and compact sets \(A,B,C \in \mathcal{C}(X)\) the Demyanov-Difference has the following properties:
  • (D1)  If \(A = B+C\), then \(C = A\stackrel{D}{-}B\).
  • (D2)  \((A\stackrel{D}{-}B)+B \supset A\).
  • (D3)  If \(B \subset A\), then \(0 \in A\stackrel{D}{-}B\).
  • (D4)  \((A\stackrel{D}{-}B) = -(B\stackrel{D}{-}A)\)
  • (D5)  \(A\stackrel{D}{-}C \subset (A\stackrel{D}{-}B)+(B\stackrel{D}{-}C)\).
In the proofs we use a new technique which is based on the following lemma.

Let \(X\) be a Hausdorff topological vector space, \(A\) be closed convex, \(B\) bounded subset of \(X\). Then for every bounded subset M we have \(\displaystyle{\overline{A+M} = \bigcap_{C\in\mathcal{E}_{A,B}}\overline{B+C+M}.} \)

We also give connections between Minkowski subtraction and the union of upper differences.

Let \(X\) be a Hausdorff topological vector space, \(A\) be closed convex, \(B\) bounded subset of \(X\). Then \(A\dot{-}B = \bigcap\mathcal{E}_{A,B}\) where \(A\dot{-}B = \{x \in X | B+x \subset A\}\).

We show that in the case of normed spaces the Demyanov difference coincides with classical definitions of Demyanov subtraction.

References
  1. Grzybowski, J., Pallaschke, D., and Urbanski, R.; Demyanov difference in infinite dimensional spaces , Constructive Nonsmooth Analysis and Related Topics, Springer Optimization and Its Applications 87 (2014), 13-24.
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