Abstract
For convex bodies inE d (d ≥ 3) with diameter 2 we consider inequalitiesW i − βW d−1 +(β - 1) W d ≤ 0 (i = 0, ⋯, d − 2) whereW j are the quermassintegrals. In addition, for a ball, equality is attained for a body of revolution for which the elementary symmetric functions d−1−i of main curvature radii is constant. The inequality is actually proved fori = d − 2 by means of Weierstrass's fundamental theorem of the calculus of variations.
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Dedicated to Professor Otto Haupt with best wishes on his 100th birthday
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Heil, E. Extensions of an inequality of Bonnesen toD-dimensional space and curvature conditions for convex bodies. Aeq. Math. 34, 35–60 (1987). https://doi.org/10.1007/BF01840122
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DOI: https://doi.org/10.1007/BF01840122