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Two optimization problems for convex bodies in the \(n\)-dimensional space. (English) Zbl 1074.52004
The paper gives upper estimates of the volume and the surface area of \(n\)-dimensional convex bodies with given diameter \(d\) and minimal width \(\omega\). The estimates are attained for the symmetric slice \(S(\omega, d)\) of the ball of diameter \(d\) bounded by two parallel hyperplanes at distance apart \(\omega\). As a corollary, the volume and the surface area of \(n\)-dimensional convex bodies with given circumradius \(R\) and minimal width \(\omega\) are estimated from above. These estimates are attained for the slice \(S(\omega, 2R)\).

MSC:
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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