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It looks to me like the authors just wanted to put in some ridiculous values which "are clearly sufficient" so that they don't have to work out the details. The choice $\eta = c_1^{1/10}$ should ensur …

This is a follow up to this question, where the optimal constant was left open. Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let \begin{equation} \mathcal{H} := \{f : P \rightarrow \m …

It doesn't hold. Note that if one chooses $P= \frac{1}{n^2} \delta_n + \frac{n^2-1}{n^2} \delta_x$ with $x$ chosen such that $\mu = 1$, i.e. $x = \frac{n^2-n}{n^2-1}$, then $\mu_3 \sim n$ asymptotical …

Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be infinitely often continuously differentiable and convex. For $d = 1$, we know that for any interval $[a, b]$, it holds for $x, y \in [a, b]$ that $$ (f …