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One functional-analytical idea by Alexandrov in convex geometry. (English) Zbl 1049.52008

The functional-analytical approach by A. D. Alexandrov is discussed to the Minkowski and Blaschke structures making the set of convex figures into a vector space. The resultant analytical possibilities are illustrated by the isoperimetric type problems of finding convex figures separated by current hyperplanes similar to the Urysohn and double bubble problems.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
49Q20 Variational problems in a geometric measure-theoretic setting
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