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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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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1answer
19 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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23 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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20 views

Subgradient of a specific norm

Suppose I have vectors $x, x_1 , x_2 \in \mathbb{R}^p$ . Let $f(x) = \| \mbox{ }\|x - x_1\|_1 , \|x - x_2\|_1 \|_\infty$. How do we characterize the sub-gradient set $ \dfrac{\partial}{\partial x} ...
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1answer
45 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
48 views

How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$? [on hold]

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 $A^*=\{x\text{ in }\Bbb R^2:ax\leqslant 1\text{ for all }a\text{ in }A\}$.
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17 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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15 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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13 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
22 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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55 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
54 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
21 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
30 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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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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31 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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20 views

Simplicial cones and simplex cones

In Ewald's book Combinatorial Convexity and Algebraic Geometry, we find the following definition. 1.8 Definition. A cone $\sigma=\operatorname{pos}\{x_1,\dots,x_k\}$ is called simple or a simplex ...
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30 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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17 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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23 views

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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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
16 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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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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27 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
24 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 ...
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1answer
38 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
21 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
27 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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37 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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2answers
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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20 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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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
20 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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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
74 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
36 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 ...