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Moderation of convex bodies. (English) Zbl 1235.52008

The theory of moderation for points of a convex body is started in this paper. It is intended as an opposite to the theory of extreme points of a convex body. If \(K\subset \mathbb{R}^d\), \(d\geq 2\), is a convex body and \(x\in K\) then the moderation number of \(x\), denoted by \(m(x)\), is computed as the infimum of the set of diameters of all caps of \(K\) (i. e. non-empty intersection between \(K\) and closed half-space) containing \(x\), divided by \(\mathrm{diam} K\). The number \(m_K = \sup_{x\in K} m(x)\) is called the moderation of \(K\). The extreme points of \(K\) are characterized by null moderation numbers. Some basic properties of moderation numbers of a convex body and its elements are the foundation of a promising direction of research.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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