theorem in dimension $n+1$, which leads to the conclusion that $\theta \mapsto |\theta| M_f(t, \theta / |\theta|)^{-1}$ is a norm on $\RR^n$ for any fixed $t \geq 0$.

Techniques from Milman’s proof of Dvoretzky’s theorem {#section3}
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It is well-known that for large $n$, the uniform probability measure $\sigma_{n-1}$ on the unit sphere $S^{n-1}$ satisfies strong concentration inequalities. This concentration of measure phenomenon is one of the main driving forces in high-dimensional convex geometry, as was first demonstrated by Milman in his proof of Dvoretzky’s theorem (see [@mil70] or [@GM Section 4.2]). Our next proposition is essentially taken from Milman’s work, though the precise formulation we use is due to Gordon [@gordon1], [@gordon2] (see also [@schechtman1], [@schechtman2] or [@NR Theorem 6]).

Let $n \geq 1$ be an integer, let $L > 0$, $0 < \eps \leq 1/2$, and let $g: S^{n-1}
\rightarrow \RR$ be an $L$-Lipshitz function. Denote $M =
\int_{S^{n-1}} g(x) d \sigma_{n-1}(x)$. Assume that $1 \leq k \leq
\hat{c} \eps^2 n$ is an integer. Suppose that $E \in G_{n,k}$ is a random subspace, i.e., $E$ is distributed according to the probability measure $\sigma_{n,k}$ on $G_{n,k}$. Then, with probability greater than $1 - \exp \left(- c \eps^2 n
\right)$, 