and Dvoretzky’s theorem are apparent from the proof of Theorem \[thm\_multi\] below. We currently do not know whether there exists a single subspace that satisfies both the conclusion of Theorem \[thm\_multi\] and the conclusion of Dvoretzky’s theorem simultaneously; both theorems show that a “random subspace” works with large probability, but with respect to different Euclidean structures. The logarithmic dependence on the dimension is known to be tight in Milman’s form of Dvoretzky’s theorem. However, we have no reason to believe that the quantitative estimates in Theorem \[thm\_multi\] are the best possible.

There are several mathematical articles where Theorem \[thm\_basic\] is explicitly conjectured. Brehm and Voigt suggest Theorem \[thm\_basic\] as a conjecture in [@brehm_voigt], where they write that this conjecture appears to be “known among specialists”. Anttila, Ball and Perissinaki formulated the same conjecture in [@ABP], independently and almost simultaneously with Brehm and Voigt. Anttila, Ball and Perissinaki also proved the conjecture for the case of uniform distributions on convex sets whose modulus of convexity and diameter satisfy certain quantitative assumptions. Gromov wrote a remark in [@gromov Section 1.2] that seems related to Theorem \[thm\_basic\] and Theorem \[thm\_multi\], especially in view of the techniques we use here. Following [@ABP] and [@brehm_voigt], 