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### How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
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### Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
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### Projecting on a convex compact polytope with special form

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...
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### Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron $$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$ Notice that $\mathcal P_X$ is symmetric about the origin. ...
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### Finding an unfrustrated set of local linear constraints with given minimal value

Let $\{F_{i}\}$ be a finite set of linear functionals on a convex compact subset $B\subset\mathbb{R}^{n}$ such that each $F_i$ is $k$-local (acts on $k\ll n$ variables only). Assume $F=\sum_{i}F_{i}$...
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### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. ...
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### Self-concordant function for dual cone

I wonder if there is any existing result for self-concordant function in the literature about the following question. Suppose $f$ is a self-concordant barrier function of a proper cone $K$ (pointed, ...
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### Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
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### Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
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Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, \... 0answers 77 views ### Linear projections of convex sets with unique preimages of boundary points Fix a compact convex subset C \subset \mathbb{R}^n with nonempty interior. For any subspace S \subset \mathbb{R}^n, let P_S denote the orthogonal linear projection onto S. I'd like to claim ... 0answers 72 views ### Concavity of mixed volumes and mixed discriminants For n\times n symmetric matrices A_1, \ldots, A_n, the mixed discriminant D(A_1, \ldots, A_n) can be defined as 1/n! times the coefficient of t_1\ldots t_n in the homogeneous polynomial \... 1answer 306 views ### Convexity of the product of two exponential matrices Let S\subset\mathbb{R} be a convex set and \mathbb{S}^{n} be the set of real symmetric matrices of order n\times n. A matrix valued function \Gamma: S \rightarrow \mathbb{S}^{n} is said to ... 2answers 404 views ### A question involving Mazur's Lemma Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): "Assume (x_n) converges weakly to x. Then there exists a sequence (y_n) made up of convex combinations of the x_n's that ... 1answer 126 views ### Takahashi convex metric spaces A Takahashi convex metric space is a metric space (X,d) such that \exists W : X \times X \times [0,1] \rightarrow X that satisfies : d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) d(u,... 1answer 112 views ### Gradient estimate of convex functions Consider a special type of convex function g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\} such that g(x)=+\infty as |x|\to \infty. Then g is differentiable almost everywhere within its ... 1answer 224 views ### monotone parabolic systems, convex variational structure and Legendre transform The context: for my research I am currently looking at parabolic systems of the type$$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
from generalized inequality, we now that for $p>q$, we have $M_p(\mathbf{x})\ge M_q(\mathbf{x})$. now I am curious to know if we can find a constant $\alpha(p,q)$ which is only function of $p,q$ ...