Central sections of a convex body with ellipsoid of maximal volume $$B_2^n$$ B 2 n
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Mathematische Annalen
Central sections of a convex body with ellipsoid of maximal volume B2n Eleftherios Markesinis1 Received: 7 May 2019 / Revised: 8 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let K be a convex body in Rn with ellipsoid of maximal volume B2n . We prove √
that every k-dimensional central section of K has volume at most
√ k+1 k+1 n √ . k! k
In
k 2
the centrally symmetric case the upper bound is ( 4n k ) . As an application of these inequalities we get extremal properties of cube and simplex. Mathematics Subject Classification Primary 52A23; Secondary 46B06 · 52A40
1 Introduction The aim of this note is to give an upper bound for the volume of k-dimensional central sections of a convex body, the only assumption is that the convex body has ellipsoid of maximal volume the Euclidean unit ball. The question of estimating the central slices have been considered for various convex bodies. An important example is the unit cube intersected with central hyperplanes, bounds for these sections have been given by Hensley [7] resp. Ball [1]. Modified questions for l p -balls have been answered by Koldobsky [9], Meyer and Pajor [11]. In the non symmetric case with simplex instead of cube sharp bounds for the volume of k-dimensional slices of the regular simplex seems to be not known, Webb has given bound for (n − 1)-dimensional sections [12] and recently Dirksen [6] gave a bound for k-dimensional sections. Fourier Analysis has been used by many authors, we refer to the book of Koldobsky [10] for related topics and recent developments in this area. The bounds given in this note are sharp and surprisingly recover important facts from F. John’s theorem and results of K. Ball that lead to inverse isoperimetric inequality,
Communicated by Jean-Yves Welschinger.
B 1
Eleftherios Markesinis [email protected] Department of Physics, University of Athens, Panepistimiopolis Zografos, 15783 Athens, Greece
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E. Markesinis
moreover fill the gap between them, which we will explain in detail in the following paragraphs. In 1948, Fritz John proved that each convex body in Rn contains an unique ellipsoid of maximal volume (see [3,8]). Thus, each convex body has an affine image whose ellipsoid of maximal volume is the Euclidean unit ball, B2n . John characterised these affine images with the following theorem which has been used many times in the theory of finite dimensional normed spaces. Theorem 1.1 The Euclidean ball is the ellipsoid of maximal volume contained in the convex body K ⊂ Rn if and only if B2n ⊂ K and, for some m ≥ n there are Euclidean unit vectors u 1 , . . . , u m , on the boundary of K and positive numbers a1 , . . . , am , for which m m a j u j = 0 and In = aju j ⊗ u j, (1.1) j=1
j=1
where In denotes the identity on
Rn .
The u i ’s of the theorem are points of contact of the unit sphere S n−1 with the boundary of K . One of the motivations for John’s Theorem is that K is contained in n B2n , the Euclidean ball of radius n, this implies that if
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