Estimates for the extremal sections of -balls

The problem of finding the maximal hyperplane section of B p n , where p > 2 , has been open for a long time. It is known that the answer depends on both p and n. In this paper, using the well-known equivalence between hyperplane sections and the isotropic constant of a body, we give an upper bound estimate for the volume of hyperplane sections of normalized ℓ p n -balls that does not depend on n and p. In addition, on the basis of results of Meyer, Pajor and Schmuckenschläger, we show further the corresponding extremal body and hyperplane section when this volume attains its minimum.