# Questions tagged [convex-analysis]

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### Does uniform convergence of suitable functions yield pathwise convergence of their convex envelopes?

For each $k\ge 1$, let $f_k:\mathbb R\to\mathbb R_+$ be $1-$Lipschitz, increasing such that $f_k(x)\ge x^+$ for $x\in\mathbb R$, $f_k(-\infty)=0$ and $$\lim_{x\to+\infty} \big(f_k(x)-x\big)=0.$$ ...
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### An inequality in an Euclidean space

For $n\geq 1$, endow $\mathbb{R}^n$ with the usual scalar product. Let $u=(1,1,\dots,1)\in\mathbb{R}^n$, $v\in {]0,+\infty[^n}$ and denote by $p_{u^\perp}$ and $p_{v^\perp}$ the orthographic ...
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### Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality: If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
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### Do separable cubic constraint and separable quartic constraint SOCP presentable?

I am an engineer who is doing some network modeling and optimization. During my work, I was running into a case that is quite strange. The problem that I am trying to solve seems to be convex and it ...
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### A continuous version of Carathéodory's convex hull theorem

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
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### Optimality condition for strongly convex function under sparsity constraint

Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...
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### Is it true that the set $\{x\colon \partial\phi(x)\subset B_r(0)\}$ is convex when $\phi$ is convex?

Let $\phi\colon \mathbb{R}^n\to \mathbb{R}$ be convex. Is it true that the sets $$A_r = \{x\in \mathbb{R}^n\;\colon\; \partial \phi(x) \subset B_r(0)\},\quad r>0,$$ are convex? For $n=1$ this ...
1 vote
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Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by $$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)... • 299 4 votes 1 answer 173 views ### Convex hull of 3 points in Cartan-Hadamard manifolds Can the convex hull of 3 points in a Cartan-Hadamard manifold be smooth? A Cartan-Hadamard manifold M is a complete simply connected manifold with nonpositive curvature (so it includes the ... • 7,087 1 vote 1 answer 72 views ### Is this notion of being "fully" convex closed under set addition? While reading through "Linear Operators: General theory" by "Jacob T. Schwartz", reading the corollary to II.10.1 which states that for a compact convex subset C of some ... • 204 1 vote 1 answer 90 views ### If |P|<\infty and C=P\cap\partial(\textrm{Conv}(P)), then P\subset\textrm{Conv}(C)? That is, if P is a finite set, and C is the set of points in P which lie on the boundary of the convex hull of P, then is P contained in the convex hull of C? It seems true intuitively. In ... 2 votes 1 answer 166 views ### Lipschitz smooth convex extension Assume that convex f: S \to \mathbb{R} with L-Lipschitz continuous gradient on some convex compact S \subset \mathbb{R}^d is given. It would be very helpful if there existed function F such ... 3 votes 1 answer 147 views ### Is a compact set of extreme points contained in a compact face? I have run into the following question in convex analysis, which I haven't found answered in the literature: Suppose that K is a "nice-enough" non-compact convex subset of a Hausdorff ... • 135 2 votes 1 answer 223 views ### Boundary points in \overline{\operatorname{conv}\{z_i\}_{i\in I}} Let X be an infinitely-dimensional Banach space and \{z_i\}_{i\in I} be a set of linearly independent points in X_{\leq 1}, the closed unit ball of X. I the index set is not necessarily ... 5 votes 1 answer 413 views ### Is there a Borel measurable f:\mathbb{R}^d \to \mathbb{R}^d such that f(x) \in \partial \varphi (x) for all x? Let \varphi: \mathbb{R}^d \to \mathbb{R} be a convex function. The subdifferential of f at x is defined as$$ \partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
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I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...