# Tagged Questions

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### Minimax-like theorems involving union and intersection of regions in $\mathbb R^d$

From Minimax theorems, we roughly know that if $f(x,y)$ is convex on $X$ and concave $Y$ for both compact $X,Y$, then: $$\max_{x\in X}\min_{y\in Y} f(x,y)=\min_{y\in Y}\max_{x\in X} f(x,y).$$ We can ...
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### Convex functions with non-singular hessian measure are continuously differentiable?

It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...
Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by $$h_K(u) = \sup \{ \sum_{i=1}^d x_i ... 2answers 143 views ### Perimeter of a 'trapped' convex set Consider the following setup: three bounded, 'nice' convex sets A \subseteq B \subsetneq C \subset \mathbb{R}^2, and three points x,y,z\in \partial A\cap \partial B\cap \partial C (see edit ... 0answers 170 views ### How large are the smallest-area projections of a high-dimensional convex body? Let B be a convex body in \mathbb{R}^d, equipped with its standard Euclidean form, and assume that$$\intop_B x \, dx = 0\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$a ... 2answers 161 views ### The epigraph of a semi-convex function has positive reach I've been trying to prove the following theorem for several hours with no result so far. Claim. Let f:\mathbb{R} \to \mathbb{R} be a semi-convex function, i.e. there exists a constant C > 0 ... 1answer 101 views ### Relative interior and dense subsets (This is a cross-post from here.) Let A,B\subseteq \mathbb R^d be non-empty, such that B\subseteq \overline A. For S\subseteq\mathbb R^d define the relative interior of S by ... 0answers 100 views ### Uniform convergence of difference quotients of a convex function Let f(\cdot):\mathbb{R}^n\rightarrow \mathbb{R} be a convex function. At x denote the subdifferential by \partial f(x) which is compact and closed. Now, define the approximation of f around a ... 2answers 147 views ### What is the dual of an semidefinitely representable (SDR) cone? The Question Let V\simeq \mathbb{R}^r be an r-dimensional vector space with the usual Euclidean inner product. Let \mathcal K\subset V be a cone defined as$$ \mathcal K=\Big\{x\in ...
Hello everyone my question is: $Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the ...