This question comes from a lemma (whose proof is left as an exercise) which I came across when reading the continuous exponential-weighting algorithm.
Let $\mathcal K\subset\mathbb R^d$ be a compact convex set with finite volume: $|\mathcal K|<\infty$. Let $u\in\mathbb R^d$ be a fixed vector and define $x^*\in\arg\min_{x\in\mathcal K}\langle x, u\rangle$. Prove that $$ -\log\left(\frac{1}{|\mathcal K|}\int_{\mathcal K}e^{-\langle x-x^*, u\rangle}dx\right)\leq 1+\max\left(0, d\log\left(\sup_{x,y\in\mathcal K}\langle x-y, u\rangle\right)\right). $$
Can someone give me some hint how to prove this inequality? I feel confused how $d$ appears in the bound. Thanks!