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On successive minima-type inequalities for the polar of a convex body. (English) Zbl 1429.52009

The authors study bounds on the volume of a convex body in terms of the successive minima of its polar body, starting from the conjectures and the estimates of K. Mahler [Čas. Mat. Fys. 68, 93–102 (1939; Zbl 0021.10403); Bull. Aust. Math. Soc. 11, 123–131 (1974; Zbl 0277.10022)] and L. Fejes Tóth and E. Makai jun. [Stud. Sci. Math. Hung. 9, 191–193 (1975; Zbl 0299.52010)]. Successive minima of the polar of difference body of triangles from a special class are computed first. Some basic properties of gauge functions and support functions are presented. Finally, two theorems establishing new upper bounds and lower bounds are proved.

MSC:

52A38 Length, area, volume and convex sets (aspects of convex geometry)
52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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References:

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