We present a simple proof with sharp estimates of Dvoretzky's theorem on the existence of almost spherical sections having large dimension in arbitrary convex bodies in $R^N$.
Yehoram Gordon. "Gaussian Processes and Almost Spherical Sections of Convex Bodies." Ann. Probab. 16 (1) 180 - 188, January, 1988. https://doi.org/10.1214/aop/1176991893