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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Explain 1 step for linear function $C$?

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
37 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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1answer
15 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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28 views

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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1answer
28 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
25 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
45 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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2answers
33 views

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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1answer
27 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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1answer
19 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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2answers
49 views

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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50 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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1answer
34 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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0answers
9 views

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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1answer
24 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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35 views

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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16 views

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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1answer
26 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
36 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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1answer
52 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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1answer
16 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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2answers
80 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
21 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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16 views

Let $x \in \mathbb{R}^N$, $Y \in \mathbb{S}_+^N$, is $x^TYx$ convex?

Let $x \in \mathbb{R}^N$ is a vector, and $Y \in \mathbb{S}^N_+$ is a positive definite matrix Is $f(x,Y)=x^TYx$ convex over the space $\mathbb{R}^N \times \mathbb{S}^N_+$?
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1answer
16 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
23 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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1answer
24 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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2answers
58 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
57 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
22 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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1answer
32 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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1answer
61 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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1answer
33 views

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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1answer
32 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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18 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
18 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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1answer
23 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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1answer
35 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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2answers
37 views

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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Proof of sufficient condition of existence of Lagrange multipliers

Consider the optimization problem $$ (P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \} $$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and $0 \ne ...
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1answer
32 views

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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1answer
18 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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14 views

mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpniski's theorem from which we can deduce that for ...
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1answer
16 views

Prove convexity of log modified bessel function

I need to prove that the modified bessel function of the second kind is log convex in the square of the argument. Specifically I'm interested in showing, $\log \mathcal{K}_0(\sqrt{x})$ (zero order) is ...
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1answer
28 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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1answer
27 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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1answer
55 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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1answer
9 views

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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3answers
26 views

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 ...