On the Derivatives of X-ray Functions
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چکیده
We show that for every n ≥ 4, 0 ≤ k ≤ n − 3, p ∈ (0, 3] and every origin-symmetric convex body K in Rn, the function ‖x‖−k 2 ‖x‖ −n+k+p K represents a positive definite distribution on Rn, where ‖·‖2 is the Euclidean norm and ‖·‖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 ≤ m ≤ n. We also prove a sharp lower estimate for the maximal derivative of X-ray functions of the order (n− 4) at zero.
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تاریخ انتشار 2002