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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Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube

Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let ...
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132 views

How I can find an optimal solution for a model with concave-convex objective function?

My objective function is sum of three functions, 2 linear functions and a concave function ($1-\exp(x)$); constraints of my model are convex. How can I obtain optimal solution from this problem?
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1answer
202 views

Is concave quadratic + linear a concave function?

Basic question about convexity/concavity: Is the difference of a concave quadratic function of a matrix $X$ given by f(X) and a linear function l(X), a concave function? i.e, is f(X)-l(X) concave? ...
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1answer
923 views

Quadratic bounds on a convex function

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a twice-differentiable convex function. It can shown be that for any $x_0 \in \mathbb{R}$: $f(x) \leq f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2}{2}$C ...
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588 views

Intersection of two (specific) convex functions

Given are the following two functions: $$g(z) = \left(z-2\right)\left(2+z\left(z-2\right)\right)$$ and $$h(z) = 2\left(z-1\right)^{2}\ln\left(z-1\right),$$ where $z>2$. I would like to show ...
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1answer
59 views

What is the connection between strong norms and norms coming from scalar products (in pre-hilbert spaces)?

In the best-approximation problem of seperation theorems in convex analysis, there is the notion of a "strong norm", in the sense that If $\| x^1 + x^2 \| = \| x^1 \| + \|x^2 \| $, $x_1 , x_2 ...
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1answer
166 views

Convex hull approximated from inside by only finite number of elements?

In approximating the convex hull "from inside", i.e. $$ \text{conv}S = \{ x \in \mathbb{R}^n \mid x= \sum_{i=1}^k \lambda_i x^i, x^i \in S, \lambda_i \geq 0, \sum_{i=1}^k \lambda_i= 1 \} ...
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1answer
93 views

Proving convexity of this set in $\ell^2$

This is a follow-up to the question I posted earlier this week. Consider, for a fixed sequence $(a_n)_n\in\ell^2$ the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all ...
4
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1answer
297 views

Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: $$A+C\subset B+C\implies ...
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74 views

Find a vector such that its matrix product is positive in every element

Given a matrix $A$ I want to find a vector $\vec{x}$ such that every element of $A\vec{x}$ is strictly positive. Also, the columns of $A$ do not span the full space, so if I were to just naively pick ...
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1answer
116 views

Intersection of 2 $p$-simplices is a finite union of some $p$-simplices

I'm looking for a non-painful proof of this assertion. A p-simplex is defined as the set of all sums $\sum_{i=0}^p t_i x_i$ with $0\leq t_i\leq 1$, $\sum_{i=0}^p t_i=1$ for a geometrically ...
2
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1answer
264 views

Convex function on Banach space

Let $(Y,\|\cdot\|)$ a Banach space and $b\colon Y\to \mathbb{R}$ a nonnegative convex function such that, for some $\mathcal{E}>0$, the set $\{y\in Y\,:\, b(y)<\mathcal{E}\}$ is nonempty and ...
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3answers
112 views

Integral of exponential function

Consider $f$ being a measurable function on $R^n$ such that $$\int_{E} e^{|f|}=1$$ ($E$ measurable) and $f$ vanishes outside $E$ . Then $f\in L^p(R^n)$ for all $p\in (0,\infty)$. I tried using that ...
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97 views

Geometry of log-concave density functions and its distribution

Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is log concave (density function). Consider now the antiderivative (distribution function) $F(t)=\int_{{ x\le t}}f(x)dx$, which is also log concave. We ...
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1answer
104 views

Where to find information on the Hilbert cube in $\ell^2$

The Hilbert cube $H$ in $\ell^2=\ell^2(\mathbb{R})$ is the subset given by $$H=\lbrace(x_n)=(x_1,x_2,\ldots)\in\ell^2:|x_n|\le2^{-n} \text{ for }n=1,2,\ldots\rbrace.$$ I've heard that ...
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2answers
89 views

Condition for convexity

Let $f:[a,b]\rightarrow \mathbb R$ be a continuous function such that $$\frac{f(x) - f(a)}{x-a}$$ is an increasing function of $x\in [a,b]$. Is $f$ necessarily convex? What if we also assume that ...
5
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2answers
193 views

Are countable intersections of convex sets convex?

Let $X$ be a Banach space $\{C_n\colon n\in\mathbb N\}$ a collection ofconvex sets in $X$. Is the set $$C=\bigcap_{n\in\mathbb N}C_n$$ convex?
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1answer
151 views

Is the following function convex?

Consider $1\le p,q < \infty , t\in \mathbb R , f,g>0$ then is $g(x)^q[f^pg^{-q} (x)]^t$ convex in the following context? Thanks for help .
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194 views

convergence of infimum

I have a question that I encountered during my internship: Consider a convergent sequence of continuous, convex functions $\{f_n(x)\}_n$ defined in $\mathbb{R}^M$. These functions are uniformly ...
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201 views

$p$-norm is a convex function of $p$

For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function $$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$$ a convex function of $p$ on $(a,b)$ for all $f\in ...
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283 views

what is the motivation of proximal mapping?

For a convex function $h$, its proximal operator is defined as: $$prox_h(x)=argmin_uh(u)+\frac{1}{2}\|u-x\|^2$$ Can anyone provide an intuitive explanation/motivation of proximal mapping?
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Boyd & Vandenberghe, question 2.31(d). Stuck on simple problem regarding interior of a dual cone.

In Boyd & Vandenberghe's "Convex Optimization", question 2.31(d) asks to prove that the interior of the dual cone $K^*$ is equal to (1) $\text{int } K^* = \{ z \mid z^\top x > 0 $ for all $ x ...
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2answers
61 views

How smooth is the distribution function of a convex polynomial?

Here is a prototype of the problem I have in mind: Let $P:\mathbb{R}^2\rightarrow\mathbb{R}$ be a strictly convex, nonnegative polynomial such that $P(0,0)=0$. Let $\alpha\geq 0$, and consider the ...
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2answers
34 views

Some equivalences for convex sets

For a subset $A\subset V$ of a vector space over $\mathbb{R}$, let $\mbox{conv}(A) := \left\{ \sum_{i=1}^n a_i x_i\, \middle| \,x_i\in A, a_i \ge 0\text{ with } \sum_{i=1}^n a_i = 1 \right\}$. I ...
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3answers
557 views

Level sets of convex functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a convex function. For $t\in\mathbb{R}$, consider the corresponding level set $$f^{-1}\{t\}=\{(x,y)\in\mathbb{R}^2: f(x,y)=t\}.$$ For the application I ...
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1answer
1k views

How to prove $f(x,y)=\sqrt {xy}$ is concave?

How can I prove (preferably elegantly) that $f(x,y)=\sqrt {xy}$ where $x≥0$ and $y≥0$ is concave in both $x$ and $y$?
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1answer
77 views

Prove that a set defined by concave functions on $R^n$ is convex

I've been trying to prove this statement the whole weekend... Let $g_1,\dots,g_m$ be concave functions on $\mathbb{R}^n$. Prove that the set $S=\{x:g_i(x)\geq{0},\ i=1,\dots,m\}$ is convex.
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182 views

Determine amount of overlap of N k-dimensional point clouds

in the area of image processing, I'm trying out different dimensionality reduction techniques. I use these to reduce intensity images/bitmaps (sliced as a vector, i.e. such a vector has a high ...
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1answer
293 views

Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$. Now my ...
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610 views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots ...
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1answer
255 views

An inequality concerning concave functions

Let $I\subset\mathbb{R}^{\ge 0}$ be an interval and let $f:I\rightarrow\mathbb{R}^{\ge 0}$ be concave (and smooth enough). I'm wondering, weather the following inequality holds: $$f(a+b) \le f(a) + ...
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1answer
409 views

On nonconvex cones over compact convex sets in Hadamard spaces

Discussion http://mathoverflow.net/questions/6627/convex-hull-in-cat0 indicates the convex hull of a finite set can fail to be closed in a complete Hadamard space. Hence the following question should ...
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66 views

Show $f(a)=\operatorname{E} \min(aX,Y)$ is a quasilinear function of $a$

Let $X$ and $Y$ be bounded real-valued random variables. Define $$ f(a)=\operatorname{E} \min(aX,Y) $$ Is $f$ a quasilinear function of $a$? That is $f$ is both quasiconvex and quasiconcave. To ...
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161 views

General quadratic form of two variables

I was referring to this lecture http://www.stanford.edu/class/ee364a/videos/video04.html. and he gave an example of a generalized quadratic equation ...
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58 views

Confusion regarding convex and affine set

I am a bit confused regarding convex and affine set. When they mention set, does it mean the set consisting of all the points belonging to the line or shape respectively?
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1answer
335 views

Hausdorff distance via support function

Let $A$ and $B$ be convex compact sets in $\mathbb{R}^n$. Define $$ h_{+}(A,B) = \inf \left\{ \varepsilon > 0 \mid A \subseteq B+\mathbb{B}_{\varepsilon} \right\} $$ where ...
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1answer
255 views

Why it is sufficient to show $|f'(z)-1|<1$?

According to an article entitled "On the Univalency of Certain Analytic Functions" by Wang et al. (2006), we have to show that $|f'(z)-1|<1$ in order to find the radius of univalency for the class ...
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569 views

I dont't understand a proof for the second-order condition for convexity

I find a proof here http://mathhelpforum.com/advanced-math-topics/129503-second-order-condition-convexity.html But I don't understand the second part. For example. What does For the converse, if ...
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1answer
136 views

Can all convex polytopes be realized with vertices on surface of convex body?

Each convex polytope $P$ has a combinatorial type, its so-called face lattice. This lattice is just the poset of all faces of $P$ ordered by inclusion. Given one realization of such a combinatorial ...
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“Round” regions on surface of convex polytope

A convex $d$-polytope $P$ is the convex hull of finitely many points. Given such a polytope with $n \gg d$ vertices, I would like to prove that its surface has to be "round" in some region. Let me ...
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58 views

To show if function $f(x), 0\leq x<\infty$ is convex then $a_{n}=f(n)$ is convex sequence.

How can we show that if the function $y=f(x)$ is convex in $0\leq x<\infty$, then the points $a_{n}=f(n)$ form a convex sequence.
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238 views

Prove one set is a convex hull of another set

Define two sets: $A = \{x \in \{0,1\}^n : \lVert x \rVert_1 \leq k\}$ is a finite set of binary vectors; $B = \{x \in [0,1]^n : \lVert x \rVert_1 \leq k\}$ is an infinite set of real-valued vectors, ...
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1answer
227 views

Smoothness and Hessian of convex conjugate/Legendre transform

Let $p:[0,\infty) \times \mathbb{R}^n \to \mathbb{R}$ be a smooth convex function for all $x \in \mathbb{R}^n$. We define its Legendre transform (or convex conjugate) as $$ p^*(t,y)=\sup_{x \in ...
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1answer
102 views

Is this function convex when the input vector is positive?

I am wondering if $f(\mathbf{x})$ is convex on the input of a vector of $n$ positive reals $\mathbf{x}$: $$f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$$ where ...
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422 views

condition for a set to be compact and convex

Is it true that a set in, say, $n$-dimensional Euclidean space is compact and convex iff its intersection with any line is empty, a single point, or a closed line segment?
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364 views

(quasi) convexity $\frac{x}{y}$

Hej, I have the function $\frac{x}{y}$ on the domain $R_{++}$. The Hessian matrix is - as I have calculated it - positive semidefinite. But I'm not really sure, if the function is really convex at ...
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1answer
210 views

Example of a separately convex function which is not rank-one convex

Can anyone give me an example of a function $f: \mathbb{R}^{n\times n} \rightarrow \mathbb{R}$, which is separately convex but not rank-one convex? By 'separately convex' I mean convexity in each ...
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91 views

Application of Anderson's theorem to probability

From Wikipedia, Anderson's theorem is stated as: Let $K$ be a convex body in n-dimensional Euclidean space $\mathbb{R}^n$ that is symmetric with respect to reflection in the origin, i.e. $K = ...
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189 views

Convex Alternatives to the Gamma Function?

The Bohr-Mollerup Theorem states that the gamma function is the unique function $f: (0, \infty )\rightarrow \mathbb{R}$ satisfying $f(1)=1,$ $f(x+1) = x f(x),$ and the condition that $\log f$ is ...
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1answer
2k views

Definition of a function being unimodal

For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal. From Wikipedia $f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ ...