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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,...