Denote by $H_{t}$ the Hopf-Lax semigroup, i.e.\begin{equation}
H_{t}f(x)=\inf_{y\in\mathbb{R}}\left\lbrace f(y)+\frac{(x-y)^{2}}{2t}\right\rbrace.\end{equation} Is $H_{t}$ negative definite on bounded, positive, continuous functions $(C^{+}_{b}(\mathbb{R}),+)$? More precisely: Is it true, that for any $n\in\mathbb{N}$, $a_{i}\in\mathbb{R}$ and $f_{i}\in C^{+}_{b}(\mathbb{R})$ we have \begin{equation}\sum_{i,j=1}^{n}a_{i}a_{j}H_{t}(f_{i}+f_{j})(x)\le0\quad\forall x\in\mathbb{R},\end{equation}
provided $\sum_{i=1}^{n}a_{i}=0$?
Although there is extensive literature on both negative definite functions and convex conjugation, infimal convolution, etc, I've not found anything on the relation between both so far. I'm grateful for any input.
Last remark: $C^{+}_{b}(\mathbb{R})$ is not so crucial, negative definiteness on any reasonable semigroup $(S,+)\subset([0,\infty)^{\mathbb{R}},+)$ would be helpful.
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