(University of Kiel)
"On volumes of sections and slabs of l_p^n -balls"
Abstract:After surveying results of Vaaler, Hensley and K. Ball as well as Oleszkiewicz and Pelczynski on minimal and maximal volume sections of the cube in R^n and C^n , we investigate formulas for the volume of slabs of the cube of width t and study directions of minimal and maximal volume. These directions turn out to be directly related to vectors attaining optimal Khintchine type inequalities for uniform variables on spheres. Similar problems for sections of unit balls of l_p^n were investigated by Meyer and Pajor, Barthe and Koldobsky, in particular, for 1<=p<2. For p>2, only very incomplete results are available. (Joint work with A. Koldobsky.) In the early 80's Lior Tzafriri and I studied Khintchine type inequalities in the context type and cotype; those estimates like the ones in the lecture were very much finite-dimensional in nature.