AN ISOMORPHIC DVORETZKYS THEOREM FOR CONVEX-BODIES

We prove that there exist constants C > 0 and 0 < lambda < 1 so that f or all convex bodies K in R-n with non-empty interior and all integers k so that 1 less than or equal to k less than or equal to lambda n/(n + 1), there exists a k-dimensional affine subspace Y of R-n satisfyin g d(Y boolean AND K,B-2(k)) less than or equal to C(1+root(k/ln(n/kln( n+1))) This formulation of Dvoretzky's theorem for large dimensional s ections is a generalization with a new proof of the result due to Milm an and Schechtman for centrally symmetric convex bodies. A sharper est imate holds for the n-dimensional simplex.