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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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