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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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