Rapid Steiner Symmetrization of Most of a Convex Body and the Slicing Problem

B. KLARTAG, V. MILMAN
2005 Combinatorics, probability & computing  
For an arbitrary n-dimensional convex body, at least almost n Steiner symmetrizations are required in order to symmetrize the body into an isomorphic ellipsoid. We say that a body T ⊂ R n is "quickly symmetrizable" if for any ε > 0 there exist only εn symmetrizations that transform T into a body which is c(ε)-isomorphic to an ellipsoid, where c(ε) depends solely on ε. In this note we ask, given a body K ⊂ R n , whether it is possible to remove a small portion of its volume and obtain a body T ⊂
more » ... d obtain a body T ⊂ K which is quickly symmetrizable? We show that this question, for a large variety of c(ε), is equivalent to the slicing problem.
doi:10.1017/s0963548305006899 fatcat:bttnnordtra3tkjhezf2jm6cxi