in Section \[section3\], and it is also outlined by Gromov [@gromov Section 1.2].

Fix a subspace $E$ such that $Proj_E(X)$ is approximately spherically-symmetric. In Section \[section4\] we use the Fourier transform to conclude that the approximation by a spherically-symmetric distribution actually holds in the stronger $L^{\infty}$-sense, after convolving with a gaussian. In Section \[section5\] we show that the gaussian convolution has only a minor effect, and we obtain a spherically-symmetric approximation to $Proj_E(X)$ in the total-variation, $L^1$-sense. Thus, we obtain a density in the subspace $E$ that has two properties: It is log-concave, by Prékopa-Leindler, and it is also approximately radial. A key observation is that such densities are necessarily very close to the uniform distribution on the sphere; this observation boils down to estimating the asymptotics of some one-dimensional integral. At this point, we further project our density, that is already known to be close to the uniform distribution on a sphere, to any lower-dimensional subspace. By Maxwell’s principle we obtain an approximately gaussian distribution in this lower-dimensional subspace. This completes the rough sketch of our proof.

Throughout this paper, unless stated otherwise, the letters $c, C, c^{\prime}, \tilde{C}$ etc. denote positive universal constants, that are not necessarily the 