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Archiv der Mathematik

, Volume 79, Issue 3, pp 216–222 | Cite as

On the derivatives of X-ray functions

  • A. Koldobsky
Article

Abstract.

We show that for every \( n \geqq 4, 0 \leqq k \leqq n - 3, p \in (0, 3] \) and every origin-symmetric convex body K in \( \mathbb{R}^n \), the function \( \parallel x \parallel^{-k}_{2} \parallel x \parallel^{-n+k+p}_{K} \) represents a positive definite distribution on \( \mathbb{R}^n \), where \( \parallel \cdot \parallel_{2} \) is the Euclidean norm and \( \parallel \cdot \parallel_{K} \) is the Minkowski functional of K. We apply this fact to prove a result of Busemann-Petty type that the inequalities for the derivatives of order (n - 4) at zero of X-ray functions of two convex bodies imply the inequalities for the volume of average m-dimensional sections of these bodies for all \( 3 \leqq m \leqq n \). We also prove a sharp lower estimate for the maximal derivative of X-ray functions of the order (n - 4) at zero.

Keywords

Convex Body Lower Estimate Euclidean Norm Maximal Derivative Definite Distribution 

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Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • A. Koldobsky
    • 1
  1. 1.Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA,¶ e-mail: koldobsk@math.missouri.eduUSA

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