An isomorphic Dvoretzky's theorem for convex bodies

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in ${\displaystyle {ℝ}^n}$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of ${\displaystyle {ℝ}^n}$ satisfying ${\displaystyle d(Y\cap K,B_2^k)\le C(1+\surd \left(\frac{k}{\ln \left(\frac{n}{k\ln (n+1)}\right)}\right)}$. This formulation of Dvoretzky's theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.