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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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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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
56 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
27 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 ...
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1answer
58 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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1answer
24 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 ...
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1answer
19 views

separate convex and concave function by affine function

Let $f:\mathbb R^n\to\mathbb R$ be a concave continuous function and $g:\mathbb R^n\to\mathbb R$ be a convex continuous function such that $f\leq g$. Then there exists an affine continuous function ...
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1answer
28 views

Dual norm equivalence?

$\|\|$ is a norm in $R^n$, its dual norm is defined as $\|s\|^*=max_{\|x\|=1}s^Tx$. We denote $s^\#$ as any vector in the following set: [Arg $max_x: \ \ s^Tx-\frac{1}{2}\|x\|^2$] How to verify ...
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38 views

Hahn-Banach separation theorem

Let $V_1,V_2$ be convex subsets of a normed space $X$ with $V_1^\circ\neq\emptyset$ and $V_1^\circ\cap V_2 =\emptyset$. Then there exists $x'\in X'\setminus\{0\}$ such that $$\operatorname{Re} ...
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39 views

function induced by optimization

Consider the following optimization $\displaystyle\max_{x_1, \ldots, x_n}\sum_{i=1}^n x_i y_i -\sum_{i=1}^n x_i\log(x_i)$ subject to $a_i\leq x_i\leq b_i$ and $\sum_{i=1}^n x_i =c$ ...
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1answer
24 views

Locally convex space - closed sets

Assume we have a locally convex topology $\tau$ induced by semi-norms $\mathcal P$, on some real vectorspace $E$. Let $\sigma$ be the locally convex topology induced by the semi-norms $$ \mathcal Q ...
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1answer
49 views

Differentiability in a neighborhood of a strictly convex continuous function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a strictly convex continuous function, and let $f$ be differentiable at the point $x_0\in \mathbb{R}$. Can we say that $f$ is differentiable on some ...
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1answer
22 views

The Dual problem of a non constraints problem?

The primal problem is $min_{w\in R^d}: P(w)$ where $P(w)=\frac{1}{n}\sum_{i=1}^n\phi_i(w^Tx_i)+\frac{\lambda}{2}||w||^2$. The dual problem is $max_{\alpha\in R^n}: D(\alpha)$ where ...
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2answers
90 views

Why am I getting a contradiction?

Let $f(x,y) = xy$ and define $$C = \{ (x,y) : xy \geq 1 \}.$$ The Hessian of this function is indefinite and has positive and negative eigenvalues. But we know the set $C$ is convex since this is ...
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1answer
75 views

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
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1answer
38 views

If we inscribed all the 6 regular convex four-dimensional polytopes in a sphere, which one would have the highest volume?

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). But what about for the 6 regular convex ...
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27 views

What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
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54 views

Let $S$ be a closed convex set & $x$ be an extreme pt of $S$ then $S-\{x\}$ is

Let $S$ be a closed convex set and $x$ be an extreme point of $S$, then $S-\{x\}$ is Convex Not Convex May or may not be convex I am thinking that the convexity doesn't fail even if we remove the ...
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15 views

Adjacency of convex hull facets

Let $C$ be a $d$-dimensional convex polytope and $p$ is a point outside of it. $C=\{f_c\}$ defined by set of facets $f_c=\{p_c,A_c\}$ where $p_c$ is a tuple of vertices and $A_c$ is a set of ...
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1answer
26 views

Example of a uniformly convex domain in $\Bbb R^n$

I am trying to understand the differences between a convex domain, and a uniformly convex domain. Intuitively, to my knowledge, a convex domain is one where any line between any two points in the ...
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1answer
116 views

Union of convex sets is convex: Strategy?

I am working in convex geometry for the summer with little experience beforehand. It's a lot of fun but it does mean I don't know some of the basic things. I know that it is not generally true that ...
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1answer
19 views

Infinite dimensional convex cones

Let $C$ be a convex cone in a topological real vector space $V$. Assume that we have a linear functional $\varphi: V \to \mathbb{R}$ such that $\varphi(x) \geq 0$ for all $x \in C$. Further assume ...
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21 views

Detecting faces of polytopes

I am working in convex geometry for the summer with little experience beforehand. It's a lot of fun but it does mean I don't know some of the basic things. I'm interested in the orbits of finite ...
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1answer
45 views

uniqueness of Hahn-Banach extension for convex dual spaces

Let $X'$ be strict convex, i.e. for all $x_1',x_2'\in X'$ with $\|x_1'\|_{X'}=\|x_2'\|_{X'}=1$ the implication $$\left\|\frac{x_1'+x_2'}{2}\right\|=1\Rightarrow x_1'=x_2'$$ holds. In this case the ...
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35 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
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Theorem about convex sets

I want to prove a theorem (the link is after the whole text here) and in order to do that I need to prove three preliminary statements. I tried to prove them all but I'm already stuck in the first, ...
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2answers
39 views

about an interesting affirmation involving convex sets

Consider the following definition Definition: Let $\Omega \subset R^n$ a bounded convex set. A point $x \in \partial \Omega$ is called an extremal point if $x$ cannot be written as linear ...
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17 views

Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
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21 views

Nominalization for being “not convex” and “not coercive”

Having a function f which is neither convex nor coercive. What is the correct nominalization for these properties? I suppose something like ...
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89 views

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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Difference quotient inequality with convex functions

A convex function on an interval $ I $ is said to be convex if for every $ 0 < t < 1 $ and $ x,y \in I $ we have that $ f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$. Prove a function is convex if and ...
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1answer
27 views

Cone of convex solutions

I have been reading a paper, on monge-ampere type equations, and the existence of a unique convex solution has been proven to exist in $C^{3,\alpha}(\Omega)\cap C^{2,\alpha}(\overline{\Omega})$, for ...
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1answer
15 views

extension of Legendre transform of a function to a larger domain

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$ and $u_0$ be a uniformly convex function define on $\Omega$. Suppose the gradient of $u_0$ maps $\Omega$ into a subdomain of $\Omega^*$, i.e. ...
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1answer
25 views

Calculating Legendre Transform

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$, $B_r(y_0)$ is a small ball of radius $r$ center at $y_0\in\Omega^*$, in $\Omega^*$ define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on ...
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2answers
35 views

Proving convexity

I ask you please some help with this problem: Let A $\subseteq$ R$^n$ be a convex set and $C(A)$ = {$\lambda$x, $\lambda \in \mathbb{R}$, $\lambda \geq 0$, x $\in$ A}. Prove that $C(A)$ is convex. ...
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2answers
53 views

Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
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1answer
41 views

Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
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Supremum of convex lipschitz functions.

Let $f_i:K\to R, i\in I$ be a family of convex, equi-Lipschitz functions on some compact subset $K$ of $\Bbb R^n$. Is it true that $\sup f_i$ is also Lipschitz continuous(assuming that the sup ...
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1answer
29 views

Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
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1answer
24 views

Is it possible to convexify this cone constraint?

General question An SOCP constraint is given by: $$ \| A_i \mathbf{x} + b_i\| \leq \mathbf{c}_i^T \mathbf{x} + d_i.$$ I have the following constraint: $$ \| A_i \mathbf{x} \| \geq d_i.$$ Is it ...
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1answer
19 views

Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$. Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in ...
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40 views

compact and convex set

I recently have worked on compact convex sets in the context of time series and my question is related to that. If we have a set $$ C=\{\beta_1 X_1 +\beta_2 X_2 +\phi H_1 , ...
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1answer
33 views

Projection onto Polyeder

I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 ...
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1answer
19 views

Series and concavity

If $u(x)$ is strictly concave, can I say: $$ \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n+1}\cdot u(n) < \infty. $$ I am having trouble finding counterexamples. Thanks.
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24 views

Convexity defined by Karamata inequality

Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?
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32 views

Separate two convex sets with disjoint interior (in $\mathbb{R}^n$)

In $\mathbb{R}^n$, I know that if $A$ is a convex set and $b$ in the boundary of $A$. Then we can separate $A$ and $b$, which means there exists $f \in \mathbb{R}^n$ such that $f\cdot x \ge f \cdot b$ ...
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1answer
17 views

Properties concave functions

Is is true that if $f(x)$ is a concave function of $x$ with domain $C$, then $f'(a) \leq \frac{f(a)}{a}$ for any $a \in C$, where $f'(a)$ denotes the derivative of $f(x)$ with respect to $x$ evaluated ...