Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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Example of convex subset (unbounded) with $\text {rec} (C) = {0}$

Example of convex subset (unbounded) with $\text {rec} (C) = {0}$ I've proved that for a bounded convex subset $C$ it always holds that $\text {rec} (C) = {0}$. However, now I'm looking for an ...
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Number of local minimums and nonconvexity

I came across the following in my reading, and I like to know why this is true. "$\dots$ but, the fuction $F:\mathbb{R}^n \to \mathbb {R}$ is nonconvex since it has several local minima $\dots$" ...
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1answer
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Lower bound on Hessian, mean-value theorem

Let $x \mapsto f(x) \in \mathcal{C}^2$ be convex, i.e. $\forall x \in \mathbb{R}^n$, $\nabla^2f(x) \succeq 0$. Let $A \in \mathbb{R}^{m \times n}$ and suppose $\nabla^2f(x) + A^\top A \succ 0$. Is it ...
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Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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What is condition for a convex polyhedron to be uniform?

A uniform polyhedron has all its vertices exactly lying on a spherical surface with a certain radius. Condition: A convex polyhedron will be uniform (i.e. all the vertices will exactly lie on a ...
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28 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...
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22 views

Definition of a fan of a polytope

In Fulton's book Introduction to Toric varieties (page 25), he says that: A rational convex polytope $K$ in $N_{\mathbb{R}}$ determines a fan $\Delta$ whose cones are the cones over proper faces ...
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K is convex.Is this true that ${K^ \circ } \ne \phi$?. [on hold]

Let $ K$ is convex.Is this true that ${K^ \circ } \ne \phi$?
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Finding the vertices of a convex set of matrices

I'm a little new here so wasn't sure if this was the right area. I've been trying to figure out how to generate a set of random $K \times N$ (for $K<N$) matrices that are subject to a several ...
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1answer
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Proximal Operator of $\ell_{\infty,1}$ norm of a matrix

How can I calculate the proximal operator of mixed norm $\ell_{\infty,1}$ for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_{\infty,1} + \frac{1}{2\tau} ||X-Y||_F^2$ where ...
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Proof of convexity of a quadratic function

I have the next problem: If $f(x)$ is a quadratic function with n variables: $f(x) = 0.5$$\mathbf{x}^T$$A$$\mathbf{x}$$+$$\mathbf{b}$$^T$$\mathbf{x}$$+$$\mathbf{c}$ were $A$ is a symmetric matrix ...
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24 views

Gluing two strongly convex function

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^n$ and $t\in[0,1]$ it follows $$f(tx+(1−t)y)\leq ...
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min max duality

I have been introduced min max duality. When i have a problem : $f:X \to R$ $(Primal) inf (f(x)):x\in A$. Let say that $A=\{x \in X: h(x)=0\}$. I can express my problem : $inf_X sup_Y L(x,y):= ...
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Weighted least squares with nuclear norm minimizaiton, how to optimize?

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} ...
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Is a convex salient cone necessarily contained in an open half-space?

A cone $C$ in $\Bbb R^n$ is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if $C \cap (-C) \subset \{0\}$. Obviously, a cone $C$ such that that ...
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Triangle Inequality Like Equation [on hold]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
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About the Affine hull and Span.

I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension ...
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Show that if A and B are strictly convex, then A + B is strictly convex or provide a counter example.

We have: If A is open: $\exists x,y \in A,$ $x \neq y$ such that $\lambda x+(1-\lambda y)\in \dot A $ (the interior) and $\exists u,v \in B,$ $x \neq y$ such that $\lambda u+(1-\lambda v)\in ...
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2answers
32 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
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1answer
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Discontinuous semiconcave functions

A function $u: \mathbb{R}^n \to \mathbb{R}$ is defined to be semiconcave if there is a positive constant $c$ such that for all $x,z$ $$ u(x-z) + u(x+z) - 2u(x) \leq c |z|^2. $$ Alternatively, one ...
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Check if ray intersects internals of $D$-facet

Given a ray $\overrightarrow{r_0} + \overrightarrow{v} \cdot t, t \in [0;+\infty)$ and a $(D - 1)$-simplex, defined by $D$-tuple of its vertices $p_i = (p_i^1, p_i^2, \dots, p_i^D), i \in \{1, 2, ...
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Discrete concavity of a log function

I want to prove that the function $f_i(P)=f_i(P_1,..P_i,..,P_K)=log(1+(\frac{a_iP_i}{\eta+\sum\limits_{i'\neq i}a_{i'}P_{i'}}))$ is discetely concave, which means that I should prove: $\forall \lambda ...
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Convergence of backtracking and gradient descent.

I am thinking a bit about the following exercise: Let $f(x) = x_1^2 + x_2^2$ with dom $f = \{ (x_1,x_2):x_1 > 0 \}$. The optimal value of this problem is $p^* =1$, but it is never attained since ...
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1answer
46 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
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Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Further, let the cdf $F$ of $\boldsymbol{X}$ be strictly increasing in each component on ...
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1answer
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Are unions, sums (…) of quasiconvex functions again quasiconvex?

for a project I need to prove quasiconvexity of several general functions. Can I argue that the union (or sum, or difference...) of quasiconvex functions is again quasiconvex? I do know that the sum ...
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31 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set ...
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Can a non-convex set be partitioned into a set of nearly convex subsets? [closed]

Consider a non-convex bounded subset $S \subseteq \mathbb{R}^{n}$. Is it always possible to partition this set into a finite set of disjoint subsets \begin{equation} S = \bigcup_{i=1}^{n}s_i, \quad ...
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1answer
19 views

Is a distance function on $\mathbb{R}^n$ convex?

Fix $z \in \mathbb{R}^n$. Let $||\cdot||$ be a norm on $\mathbb{R}^n$, and define the distance function $f(x)=||z-x||$ for $x\in \mathbb{R}^n$ Then, is it true that $f(x)$ is convex?
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Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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1answer
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How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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1answer
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Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
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Specific polygons in \R^{3} [closed]

Find all (convex) polyhedra P in $\mathbb{R}^{3}$ with following property: for every two vertices $v, u \in P$: $[u,v] \in \partial P$ . Here $[u, v] = \{w \; \vert \; w = tv + (1-t)u, \; t \in ...
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Uniformly convex and strictly convex

I have the following definitions of uniformly convex and strongly convex Let $f:R^n \to R$ be smooth. (1) $f$ is uniformly convex if there exists $\theta > 0 $ such that $$\Sigma_{i,j}f_{x_i x_j} ...
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1answer
7 views

Supporting hyperplane of convex function

Below is the appendix B of Evan's PDE book on supporting hyperplanes of convex functions. In the remark (1), he says that the mapping $y\to f(x)+r\cdot(y-x)$ determines the supporting hyperplane to ...
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Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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How can I find discrete points in convex hull

Say, I have a set of finite numbers of data $S = \{z_1, z_2,...,z_n\}$, $z_i \in \mathbb{Z}^d$, $C$ is the convex hull generated by $S$, that is, $C = conv(S)$. How can I find some discrete points in ...
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1answer
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Showing that a given function is convex.

I am trying to show that the function $f(x,\vec{y})=\alpha\ln(1+\exp(x+\vec{y}\cdot \vec{z}))+(1-\alpha)\ln(1+\exp(-x-\vec{y}\cdot\vec{z}))$ is a convex function of $(x,\vec{y})$ (where ...
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Parametrizing the Boundary of a Convex Set

Let $K$ be a compact convex set in $\mathbb{R}^2$. In the proof of a proposition in a paper I am reading, they are concerned with parameterizing $\partial K$ in the following way: If $K$ is ...
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Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
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1answer
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Strong convergence of convex combinations of a weakly convergent sequence

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 ...
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1answer
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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 (weak-* sense) set of probability measures. $m$ is a probability ...
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does convexity implies $g(u)>cu$?

So I have been doing some self study and I was wondering if my results are true, or if I am misreading something. Say we have a function $g$ which is concave on values of $u \in \mathbb{R}$. Then we ...
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1answer
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convexity and first derivative

Let $\phi$ be a differentiable function on an interval $(a,b)\subset R^1$. If $\phi '$ is non-decreasing, then $\phi$ is convex. But, is the converse true? Does the convexity of $\phi$ necessarily ...
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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) ...
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1answer
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Show the Gini Coefficient is Quasiconvex

The Gini-coefficient is defined as $$ G(x) = \sum_{i = 1}^n \frac{i}{n} - \sum_{j=1}^{i} \frac{x_{(j)}}{\mathbb{1}^{T}x}, $$ where $x_{i} $ is nonnegative numbers with positive sum. $x_{(j)}$ denotes ...
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2answers
29 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
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1answer
34 views

Show that f(x) is convex

Show that f(x) = inf{g(x1)+h(x2)} is convex subject to x1+x2=x and where g(.) and h(.) are convex functions. Can I just go about this by using the regular definition of a convex function or which ...
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0answers
38 views

Application of the Operator norm $\|.\|_O$ on the differential $df \in \hom(\mathbb{R}^n, \mathbb{R}^m)$

This question origins from my Analysis II Script which gives the following statement (without proof): Lemma Let $U \subset \mathbb{R}^n$ be convex and $f \in C^1(U, \mathbb{R}^k)$ then we have $$ ...
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1answer
23 views

Convexity/concavity of a strictly increasing and continuous function

Consider a continuous, strictly increasing function $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ with $f(0)=0$, and $x>f(x)$ for all $x>0$. Is this enough to conclude anything about ...