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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Convexity on a direction

In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have: A function $f$ defined on a triangle $T$ is said to be convex in the ...
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Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
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What is a $0$-sublevel set?

I read the notes of S. Boyd, and am confused about the following: $f_0(x)$ is quasiconvex. I am confused about the latter one particularly. What does it mean? Thanks!
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Show P is linear if it is convex and positive homogenious functional

it might be too simple but I couldnt show the second part L linear real space $P:L\rightarrow \Bbb R$ is called positive homogenious functional if for every $x\in L$ and $\alpha\ge 0$ , $P(\alpha ...
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Alternative characterization of a finite dimensional affine set

As the definition in the S. Boyd's textbook: My question is the following representation: What is the relationship between this representation and the definition above it? EX: sum of elements ...
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(Strictly) concave/convex function: Increasing / decreasing slope triangle?

I have the following (possibly quick) question. In a paper I am working with, the following conclusions are drawn which I have a hard time to understand. Since they are given without proof, I assume ...
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The super convex hull of a set in on $\Bbb R^n$

The super convex hull of a set $A \subseteq \Bbb R^n$, is the set of all $\sum_{i=1}^{\infty}\lambda_i x_i$ such that $\lambda_i \geq 0$ and $\sum_{i=1}^{\infty}\lambda_i =1$, which is denoted by ...
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Duality gap of nonconvex problem

I have an optimization problem (presumably) nonconvex but the objective funtion is increasing, continuous, and smooth. I also have a set of linear constraints which are fulfilled with equality, i.e., ...
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Find $C\subset \mathbb{R}^2$ convex unbounded such that $\vert C \vert $ is not convex?

The question is almost posted in the title and one thing to put is that $$\vert C \vert : = \big\{ ( \vert x \vert , \vert y \vert )\in [0, +\infty)^2 \, : \,\, (x, y)\in C \,\, \big\} $$ If $C$ ...
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Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
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Krein-Milman theorem

Let $E$ be a Riesz space with order unit $u$. With $$\left\|f\right\|_u: = \inf\{\lambda \in [0,\infty): -\lambda u\leq f\leq \lambda u\} $$ $E$ becomes a normed space. The following sets are subsets ...
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Representation of half-space

For any non-zero $\mathbf{y}\in\mathbb{R}^\mathcal{l}$ one half-space through origin is defined by $H_{\mathbf{y}}^{\leq}(\mathbf{0})=\left\{\mathbf{x}\in\mathbb{R}^{\mathcal{l}}:\mathbf{y}\cdot ...
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Monotonic perimeters? [duplicate]

Let C and D be two compact convex sets in the plane with respective perimeters Per(C) and Per(D) . If C is properly contained in D does it follow that ...
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Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
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Explain a linear function property?

Let $C(x)$ a linear function on $\mathbb{R}$. Then we have: $$ \begin{align*} C\left(S_0^1\right)&=C\left( \frac{y_m-r}{y_m-y_1}S_0^1+\frac{r-y_1}{y_m-y_1}S_0^1 \right) \\ ...
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Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
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taking convex hull does not add extreme points

can anybody help me please? Is there a good way to prove that given a set of points, say $S = \{x_1, x_2, ..., x_n\}$, then show that the convex hull of $S$, that is, $conv(S)$ contains all the ...
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Uniform distribution on convex hull

Let $X=\{ x_1,\dots,x_n \} \subset \mathbb{R}^m$. Let $H(X)$ be the convex hull of $X$. Assume that $X$ is a convexly independent set, i.e. none of the $x_i$ are a convex combination of the others. ...
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Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
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Difference between subspace and subset

Can you give the definition of subspace and subset of $\mathbb{R}^n$ and how can I determine their dimension?
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Discrete Cauchy integral formula : The interior values are always convex combinations of exterior values for harmonic functions?

Let $T={\mathbb Z}^2$. For $t=(x,y)\in T$, the neighborhood $N(t)$ of $t$ is the four-point set $\lbrace x\pm 1;y\pm 1\rbrace$. A map $f:T \to {\mathbb R}$ is harmonic iff $4f(t)=\sum_{s\in ...
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How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
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Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
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3D Convex hull in 3D Convex hull

I have two convex hull, How to check if the smaller one is wholly, partially or not inside the Bigger Convex hull?
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Inequality with mean value theorem with convex function [closed]

Let $f:[a,b] \rightarrow \mathbb R$ be a convex function. Prove that $$ f \left( \dfrac{a+b}{2} \right) \leq \dfrac{1}{b-a} \int_a^b f \leq \dfrac{f(a)+f(b)}{2}.$$ Any hint on how to approach this?
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Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
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Strong convexity of a function with cases

Given a set $S = \{x_1,\dotsc,x_n\} \subset \mathbb{R}$, is the function \begin{align} f&: (0,\infty) \to \mathbb{R} \\ f&(p) = 2p^2 + \frac{1}{n}\sum_{i=1}^n \max(0, -p^2-x_i) \end{align} ...
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Sub-gradient of Maimum of Multi-variable functions

The sub-differential of the maximum of a set of convex function is the convex hull of the set of active functions at that point, that is the set of functions that equal the maximum of them at that ...
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Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
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proof of the existence of spherical sections of ellipsoids

i want to prove : Let L be proper ellipsoid with the origin as center in $E^{2m-1}$ .There exists a subspace $E^m$ such that $E^m$ intersects $L$ is an m-dimensional sphere it is proven by Dvoretzky ...
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A question Kolmogrov's generalized inequality for projection onto convex sets

Kolmogrov's inequality says that, if $C$ is a convex set, and $P_C(x)$ is an operator for projecting point $x$ into the convex set $C$, if $z = P_C(x)$, then for any $y \in C$ we have $$ (z - y).(x - ...
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Why is it a borel set on the boundary of the unit ball of $E^n$?

Given $C$ convex body (compact convex set with non-empty interior points) in $E^n$ symmetric about the origin and containing the unit ball. Let $A(r)$ denote ,for every real $r >1$, the subset of ...
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what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set ...
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New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
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Question about strongly convexity and affinity

For a function $f$, it is said to be strongly convex if for all $x,y$ \begin{equation} (\nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2 \end{equation} for a constant $m \ge 0$ Is it called ...
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How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$?

How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$? If the polar dual of a set $A$ is $A^*=\{x\text{ in }\Bbb R^2:ax\leqslant 1\text{ for all }a\text{ in }A\}$. I study from ...
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Optimality conditions in convex programming

I'm reading about Zero-order conditions in Nonlinear Programming and the following confuses me (my questions are below the theory): Consider the set $\Gamma \subset E^{n+1} = \{(r,\textbf{x}): ...
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Understanding convex hull

I'm having trouble understanding the definition of convex hull. Can somebody give me an example? For example if I have the real convex set $(a,b)$ then what is its convex hull? Is it $(a,b)$ or is it ...
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Prove or disprove a statement about testing the convexity of a set using the vertices

Assume we are working in $\mathbb R^d$. Let $A=\text{Conv}(V)$, the convex hull of $V$. Also $B=\text{Conv}(W)$. I am in a situation where I can prove the following: the line segment joining $v$ and ...
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On level set of concave function

The problem is to show the following: Let $\varphi$ be a closed concave function, and $M=\max_{x \in \mathbb{R}^d} \varphi(x)$. Let $D_r:=\{\varphi\geq r\}$ be the level set. Then given $r \leq s ...
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How to approximate this large sum of exponential terms

Is there any way to approximate the following sum: $$ \sum\limits_{i_1=1}^N\sum\limits_{i_1=2}^N \cdots \sum\limits_{i_k=1}^N \cdots\sum\limits_{i_N=1}^N \exp(-r_{i_1}-r_{i_{k+1}}-r_{i_{2k+1}}- ...
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How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
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Reference for a property of convex function

Let $F:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. It is known that if $F$ has a strict local maximum, then it is not a convex function. I just would like to ask you for a ...
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Show that $L^p$ norm is logarithmically convex as a function of $p$

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded domain. Let $u : \Omega \to \mathbb{R}$ be measurable with $||u||_\infty < \infty$. For $p \in [1, \infty)$, define $$\Phi_u : p \mapsto ...
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Name of a shape that is intersected once by each ray that starts at a given point

Is there a particular name for a shape that is intersected exactly once by each ray that starts at a given point? To illustrate: I'm looking for a name for shapes like the left one in this image: ...
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Equations for interior of platon solids

It is well known that for platon solids: The interior of cube a.k.a. hexahedron can be described with equation $\max\{|x|,|y|,|z|\}<a$ The interior of octahedron - $|x|+|y|+|z|<a$. But what ...
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Show that the ellipse and the hyperbola are convex

In Spivak's chapter on differentiation, he asks the reader to prove that the tangent line to an ellipse or a hyperbola intersect the figure at exactly one point. How is this done most elegantly? I ...
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closed form solution of a particular convex program

I wish to know if there is a closed form solution of a program of the following form $\max_w x^Tw \text{ such that } \tau_2\| w \|_2 + \tau_1 \| w \|_1 \leq 1, ~\ \tau_1, \tau_2 > 0$ When either ...
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Sum Of 2 Convex is affine. Prove 2 functions are affine

Let here be two convex functions: $f(x)$ and $g(x)$ let there be two real numbers: $a$ and $b$ so it is known that: $f(x) + g(x) = ax + b$ Prove that $f(x)$ and $g(x)$ are both affine *meaning that ...
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Show convexity of $f$ in $(x,y)$

Suppose $h$ is a convex function. Let $x$ and $y$ be vectors of possibly different lengths, and $A$ a matrix. Show that the function $f$ defined as $$ f(x,y) = h(y) \qquad Ay=x\\ \qquad \qquad \infty ...