Demyanov Difference in Infinite-Dimensional Spaces

Abstract

For finitely sets (A i ) i∈I a generalization of the separation law holds and it can be shown, that a separating set can be constructed from Demyanovdifferences of the sets A i .We consider conditional minimality:It is possible to consider the problem pairs of convex sets in the more general frame of a commutative semigroup S which is ordered by a relation ≤ and which satifies the condition: if as ≤ bs for some s ∈ S, then a ≤ b. Then (a, b) ∈ S 2 = S × S corresponds to a fraction a/b ∈ S 2 and minimality to a relative prime representation of a/b ∈ S 2 .

The quasidifferential calculus, separation of convex sets and the Demyanov difference

“…Since in the finite dimensional case the exposed points are dense in the set of extreme points of a compact convex set, this definition coincides with the original definition of the Demyanov difference in finite dimensional spaces. Note that in [4] a generalization of the Demyanov difference to arbitrary topological vector spaces is also given.…”

“…Then we switch to generalized convexity and continue with the Minkowski duality. Within this context we discuss in details a separation theory for bounded closed convex sets by convex sets [3] and present a generalization of the Demyanov difference for closed bounded convex sets in arbitrary Banach spaces [4]. Here we point out in particular the very natural connection between the algebraic lattice rules and their geometric counterparts for convex sets.…”