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Let $X$ be a Hilbert space, and suppose that $f:X^2\rightarrow \mathbb{R}$ is a Lipschitz, supercoercive, convex function such that (for every $y \in X$) the set $$ \operatorname*{argmin}_{x\in X} f(x …

Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?

Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non …

I've recently come across many results discussing the differentiation of the Moreau envelope defined by \begin{equation} e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(h), \end{equation} where $f$ is …

Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator $$ \begin{aligned} &\Gamma_0\rig …

Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically infi …

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$ is $C^1$ if …