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Direct and inverse addition in convex analysis and applications. (English) Zbl 0714.46009
The author defines the direct \((A\oplus_ pB)\) and inverse \((A\square_ pB)\) addition of order \(p\in [0,\infty]\) for pairs of convex sets in a locally convex topological linear space X as follows:
Let A,B\(\subset X\) convex sets, \(0\in A\cap B\). Then \[ A\oplus_ pB=\cup \{\lambda_ 1A+\lambda_ 2B:\;\lambda \geq 0,\quad \| \lambda \|_ q=1\},\quad A\square_ pB=\cup \{\lambda_ 1A\cap \lambda_ 2B:\;\lambda \geq 0,\quad \| \lambda \|_ q=1\}, \] where \(1/p+1/q=1\) and for \(\lambda =(\lambda_ 2,\lambda_ 2)\in {\mathbb{R}}^ 2\), \(\lambda_ 1,\lambda_ 2\geq 0\), \(\| \lambda \|_ q=(\lambda^ q_ 1+\lambda^ q_ 2)^{1/2}\) if \(q\neq \infty\) and \(=\max \{\lambda_ 1,\lambda_ 2\}\) if \(q=\infty.\)
Direct and inverse addition of order p for pairs of positive functions are also defined. They are applied in the theory of second-order subdifferentials for convex functions and in the algebra of ellipsoids.
Reviewer: V.Anisiu

MSC:
46A55 Convex sets in topological linear spaces; Choquet theory
49J52 Nonsmooth analysis
46G05 Derivatives of functions in infinite-dimensional spaces
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