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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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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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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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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1answer
44 views

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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1answer
150 views

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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35 views

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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165 views

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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1answer
222 views

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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1answer
82 views

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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1answer
166 views

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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41 views

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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79 views

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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2answers
160 views

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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1answer
57 views

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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1answer
59 views

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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1answer
100 views

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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68 views

Convexity of product of two given functions

Let $f(x)$ be a convex function of $x$ in a given positive interval. Also assume $f(x)\geq 0$ everywhere in that interval. Is the function $g(x)=xf(x)$ convex in that interval?. Is it possible that ...
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1answer
35 views

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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1answer
79 views

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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Gradient-descent algorithm always converges to the closest local optima?

Assume $f(\vec x)$, which is Lipschitz continuous, has two local optima $\vec x_1^*$ and $\vec x_2^*$( $\vec x_1^*$ is the global minimum). We start the gradient-descent algorithm from $\vec x_0$ and ...
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1answer
27 views

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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56 views

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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1answer
86 views

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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1answer
53 views

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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201 views

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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1answer
39 views

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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67 views

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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108 views

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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1answer
47 views

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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123 views

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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457 views

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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65 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
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32 views

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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1answer
85 views

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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24 views

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 ...
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Existence of function $f:R^2 \rightarrow R$ s.t. f is convex in x- and y- directions and f has multiple minima.

Does there exist a function $f\colon \mathbb{R}^2 \rightarrow \mathbb{R}$ such that (1) for all $(x,y) \in \mathbb{R}^2$, f is convex in the x-direction and y-direction (2) $f$ has multiple ...
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1answer
52 views

Convexity of Norm of Max

Let $p \ge 1$. Show the convexity of the function $h:\mathbb{R}^k \rightarrow \mathbb{R}$ defined as: $$h(\textbf{z})=\left(\sum\limits_{i=1}^k \max\{z_i,0\}^p \right)^{1/p}$$
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Composition of convex and power function

Let $g$ be a convex nonegative function, and $p\ge1$. To show: $f(x)=g(x)^p$ is convex. Let $h(x)=x^p$. Then clearly $f=h \circ g$. Denote $\tilde{h}$ as the extended-value extension of $h$, which ...
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Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
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29 views

Question about relative interiors and convexity

Suppose that $C\subseteq \mathbb{R}^n$, such that $\operatorname{ri} C\neq \emptyset$ is convex and $\operatorname{cl} C$ is convex. Can we show that ...
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355 views

isoperimetric inequality using Fourier analysis

I'm trying to prove an isoperimetric inequality, but I have absolutely no idea how to go about it. let $\Gamma$ be a closed plane curve parametrized by $\gamma(t) = (x(t), y(t))$ on $[-\pi, \pi]$. ...
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Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
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120 views

The convexity of convex function's range

Given a convex function $f\colon X \to \mathbb R$ with convex domain $X \subseteq \mathbb R^n$, is the range of $f$ a convex set also?
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54 views

Help understanding an application of Jensen's inequality

This is from the book Pattern Recognition and Machine Learning by Christopher Bishop. The author states the following form of Jensen's inequality: $f\left(\int{xp(x)dx}\right) \leq \int{f(x)p(x)dx}$ ...
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written $h(t)$ versus two convex functions

given a function $h(t)$ is it possible to written it as a difference of two convex functions $h_1(t)$ and $h_2(t)$ as follow? $h(t)=h_1(t)+h_2(t)$. To clarify, every function for example $g(t)$ can ...
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Inequality for concave functions

This shouldn't be too hard, but I'm stuck. Suppose $f$ is a concave function on the interval $[a,b]$, meaning $$\lambda f(x) + (1-\lambda) f(y) \leq f(\lambda x + (1-\lambda) y)$$ for every $x,y \in ...
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1answer
189 views

Proximal Mapping for maximum of linear and quadratic function

I was wondering if there is an efficient way of calculating the proximal mapping of the following function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$, $b_i \in \mathbb{R}^3$, $c_i \in \mathbb{R}$ : $$ ...
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can any continuous function be represented as a sum of convex and concave function?

I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave. There could be ...
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1answer
54 views

Continuity of bounded and convex function on Hilbert space

I'm looking for the proof (or at least hints how to prove it) of theorem: Let $H$ be a real Hilbert space and let $f:H\to (-\infty,+\infty]$ be a convex function, bounded from above in some ...