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Maximal sections of l_p^n balls for large values of p.

Piotr Nayar
Uniwersytet Warszawski
April 27, 2023, 12:15 p.m.
room 3160
Seminar of Probability Group

Let B_p^n be the unit ball in the standard p-th norm in R^n. Suppose we intersect this ball with a codimension one hyperplane H and ask the following question: for which H is the volume of this section maximal and minimal? The first result concerning this problem is due to Hadwiger and Hensley who independently found minimal sections of the cube (case p=+oo), whereas Ball showed that H perpendicular to (1,1,0,...,0) gives the maximal section. Finite values of p were treated by Meyer and Pajor who found minimal sections for p>2 and maximal sections for 1<p<2, and by Koldobsky who found minimal sections for 1<p<2. The case of maximal sections for p>2 remained open. During the talk we shall partially answer this question by showing that Ball's direction gives the maximizer for p>10^15.