isotropic and log-concave, hence $\sup g
\leq 1$ by (\[eq\_547\]). Note that $g$ is continuous in the interior of its support, since it is a log-concave function. According to (\[eq\_621\_\]), the function $t \mapsto g(t \theta)$ is the derivative of the function $t \mapsto M_{f} (\theta, t)$, and (\[eq\_547\_\]) follows.

Our next proposition is essentially taken from Anttila, Ball and Perissinaki [@ABP], yet we use the extension to the non-even case which is a particular case of a result of Bobkov [@bobkov Proposition 3.1]. A function $g: S^{n-1} \rightarrow
\RR$ is $L$-Lipshitz, for $L > 0$, if $|g(x) - g(y)| \leq L |x-y|$ for all $x, y \in S^{n-1}$.

Let $n \geq 1$ be an integer. Let $t \in \RR$ and let $f: \RR^n \rightarrow [0, \infty)$ be an isotropic, log-concave function. Then, the function $$\theta \mapsto M_f(\theta, t) \ \ \ (\theta \in S^{n-1})$$ is $C$-Lipshitz on $S^{n-1}$. Here, $C > 0$ is a universal constant. \[lem\_bobkov\]

The proof of Proposition \[lem\_bobkov\] in [@bobkov] involves analysis of two-dimensional log-concave functions. A beautiful argument yielding Proposition \[lem\_bobkov\], for the case where $f$ is an even function, appears in [@ABP]. The approach in [@ABP] is based on an application of Busemann’s 