Remarks on Bourgain’s Problem on Slicing of Convex Bodies

Abstract

For a convex symmetric body K ⊂ ℝⁿ we define a number L _K by:

$$ nL_K^2{\left| K \right|^{2/n}} = \mathop {\min }\limits_{T \in SL(n)} \frac{1} {{\left| K \right|}} \cdot \int\limits_K {{{\left| {Tx} \right|}^2}dx\;(where\,\left| K \right| = volume\,of\,K)} $$

If the minimum is attained for T = id we say that K is in isotropic position. Any K has an affine image which is in isotropic position.