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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How to check convexity?

How can I know the function $$f(x,y)=\frac{y^2}{xy+1}$$ with $x>0$,$y>0$ is convex or not?
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1answer
316 views

Definition of an extreme set?

I have an issue with a definition in Rudin's Functional Analysis in the paragraph regarding the Krein-Milman Theorem. "Let $K$ be a subset of a vector space $X$. A nonempty set $S$ in $K$ is called ...
0
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1answer
291 views

Convex non-increasing, but not lower semicontinuous function?

I am trying to think of an example of a function $f:\unicode{x211D}^n \rightarrow \unicode{x211D}$ that is convex and non-increasing, but not lower semicontinuous, without any luck. Is this actually ...
3
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1answer
307 views

lipschitz property of the derivative of a convex function

Let $f\in C^1(\mathbb R^n\to \mathbb R)$ be a convex function. Suppose the equation $$f(x+\Delta x)-f(x)-\bigr<f'(x),\Delta x\bigl>\leq A|\Delta x|^2$$ holds for some constant $A>0$, ...
0
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1answer
70 views

convex sets , and some union of lines between two sets

Given $x,y \in \mathbb{R}^n$, let´s denote $$ [x,y] = \left\{ {u \in \mathbb{R}^n :u = tx + \left( {1 - t} \right)y\,,\,\,\,0 \leqslant t \leqslant 1} \right\} $$ Let $X , Y$ subsets of ...
3
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2answers
306 views

Convexity of inverted sum of positive definite matrices

I am currently working with a class of functions, where every function looks like $f(x)=v^T(xA+(1-x)B)^{-1}v,$ where $v\in\mathbb{R}^n$ is an arbitrary vector, $A,B \in \mathbb{R}^{n\times n}$ are ...
7
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4answers
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Is every monotone map the gradient of a convex function?

Recently in a seminar someone mentioned that monotone maps are equivalent to gradients of scalar convex functions, but it's not clear to me why this is true. One direction of the equivalence is ...
3
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1answer
323 views

Concave functions on discrete domain

We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property: $f(x+y) \leq f(x) + f(y) $ for all natural numbers $x$ and ...
0
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1answer
222 views

How to optimize entropy under linear constraints?

My problem is quite cumbersome. In general, it can be modelled as a non-linear programming problem, with linear constraints and non-linear objective function. The objective function is conditional ...
0
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1answer
55 views

If E is a subset of a lattice closed under addition then is the intersection of E with the opposite of some translate finite?

this seems intuitive to me but I'm struggling to prove it (is it false?). Let $E$ be a subset of a lattice (free abelian group of finite rank) closed under addition, containing the origin and such ...
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2answers
3k views

Is the composition of $n$ convex functions itself a convex function?

Is a set of convex functions closed under composition? I don't necessarily need a proof, but a reference would be greatly appreciated.
3
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1answer
413 views

Basic properties of the point-to-set distance function

Let $X$ be a normed vector space, $x\in X$ and $Z\subseteq X$. Then we define the point-to-set distance function as: $$ \|x\|_Z = \inf_{z\in Z} \| x-z\| $$ I use the notation $\|\cdot\|_Z$ for ...
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3answers
111 views

Reference for showing two closed and convex sets are equal

Suppose I have two sets $P, Q \subseteq R^d$ such that $P\subseteq Q$. $P$ and $Q$ are both convex and closed. I wish to show that $P=Q$. A straightfoward way to show this is showing $\forall y \in ...
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0answers
311 views

Is conditional entropy a convex function?

A conditional entropy can be expressed in the following way, $H_{V_t}(V_s) = -\sum_{s,t}p(s,t)\log{p_t(s)} = -\sum_{s,t}p(s,t)\log{\frac{p(s,t)}{\sum_{s'}{p(s',t)}}}$ $s$ and $t$ are defined ...
5
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4answers
287 views

Separation theorem

Is it true that given a real vector space $X$ and two disjoint convex sets $A,B\subseteq X$, there is always a linear functional that (weakly) separates them? I.e., is there a non-zero linear ...
2
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1answer
134 views

Generalizations of the Convex Hull

I am aware of generalizations of the convex kernel, via the addition of more polygonal line segments between points in a set. However, I wonder if there are similar generalizations for the convex ...
3
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0answers
80 views

Extreme rays of totally monotone function's cone

A smooth function $f \colon \mathbb{R}^{n}_+ \to \mathbb{R}_{+}$ is said to be totally monotone iff $(-1)^{| \alpha|} \frac{ \partial^{| \alpha |} }{\partial^{\alpha}x} f(x) \geq 0$ for any ...
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1answer
2k views

prove this is a strongly convex function

The definition of strongly convex from Wikipedia: It is not necessary for a function to be differentiable in order to be strongly convex. A third definition for a strongly convex function, with ...
2
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0answers
236 views

Accesible Area of Discrete Geometry for Undergraduate Research [closed]

This summer I will have a chance to work on a 16-week summer research project under a professor in convex/discrete geometry. I'm a first-year student with a fairly good background for my age and I've ...
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0answers
112 views

Topology of “almost convex” sets

Let $C$ be a compact set in $\mathbb{R}^n$. If $C$ is convex, it must be homeomorphic to a closed ball. Now suppose that instead of convexity we require the intersection of $C$ with any line to have ...
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1answer
87 views

Visualizing log-concavity and interlacing of roots

Someone mentioned the following fact to me today: if $\ln f(x)$ is concave, then the roots of $f(x)$ and $f'(x)$ interlace. The proof is very simple: the assumption implies that ...
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1answer
393 views

Convex Conjugate of Absolute Norm

Let $f:\mathbb{R}\rightarrow[-\infty,\infty]$ be a continuous function. The convex conjugate of $f$ is: $$f^*(p) := \sup_{x\in\mathbb{R}}\{px-f(x)\}~.$$ Furthermore, let us define the subderivative ...
2
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1answer
239 views

A question about the coercivity of a lsc and convex function.

I was doing a proof and I need to show a result to conclude it: $X$ is a reflexive Banach space with a norm, $\|\cdot\|$, of class $\mathcal{C}^1$. $f:X\to\overline{\mathbb{R}}$ is lower ...
2
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0answers
181 views

Convexity of the set of gradients of a convex function

If $\vec{y}_1$ and $\vec{y}_2$ are the gradients of a differentiable convex function $f(x)$ at points $\vec{x}_1$ and $\vec{x}_2$ does there exist a $\nabla f^{-1}( \alpha \vec{y}_1 + ...
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0answers
64 views

Relaxing strict convexity

Suppose $F: \mathbb{R^2} \rightarrow \mathbb{R}$ is convex. If $F$ is moreover strictly convex, then $$ \frac{1}{2} F(x_1,y_1) + \frac{1}{2} F(x_2, y_2) - F\left( \frac{x_1 + x_2}{2}, \frac{y_1 + ...
3
votes
1answer
389 views

Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
3
votes
1answer
229 views

Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
3
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0answers
72 views

Using convexity and separation to prove bounds on norm bound functionals

I'm quoting here a homework problem with two clauses. I've already managed to find a solution for the first clause, and have problems generalizing it for the second clause, I'll go into details after ...
11
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1answer
624 views

Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$

For $a,b \in \mathbb R$, $p\geq2$ I try to show $$\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p.$$ Is this a popular inequality (At least I could not ...
3
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1answer
355 views

Two questions regarding the ergodic decomposition theorem

In Walters' An Introduction to Ergodic Theory, page 153, Remark (2), he writes If $E(X,T)$ denotes the set of extreme points of $M(X,T)$ then for each $\mu \in M(X,T)$ there is a unique measure ...
3
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0answers
61 views

What proofs are based/use/depend on existance convex hull or convex envelope?

I wonder if there are any is at least one proof of any mathmatical fact that would use convex hull or convex envelope as mathmatical object valuable for that proof?
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1answer
274 views

How to show it is convex?

From a journal entitled Certain subclass of starlike functions by Gao and Zhou in 2007, they mentioned that " since $ k(z)=\frac{z}{1-zt}$ is convex in open unit disk $E,z:|z|<1$, $k(\bar{z})= ...
2
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2answers
368 views

Originality of Research [closed]

How can I discover the originality of a field of research or a sub-field of research without divulging the particulars? I am proving some interesting results in a potentially new sub-field of convex ...
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0answers
102 views

Existence of 'Lower bound' functions for convex functions on radially open convex sets

I am trying to solve a problem which is as follows: Suppose $f$ is a convex function on a radially open convex subset $C$ of a vector space $E$, and $x \in C$. Show that there exists a linear ...
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0answers
134 views

Panel structure on epi F**

It is known that, generically, the convex hull of a hypersurface embedded in $\mathbb{R}^n$ has a panel structure of simplices. Of course one can construct embeddings where this is not the case, but ...
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0answers
461 views

Is every convex-linear map an affine map?

Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. ...
0
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2answers
198 views

I need help showing this proof of convexity

Let $X$ be a nonempty convex subet of $A$. I need to Show that $z$ is an extreme point of $X$ if and only if the set $X − \{z\}$ is a convex set.
4
votes
2answers
113 views

Equality in definition of midconvex function

Let $f:I \rightarrow \mathbb{R}$, where $I\subset \mathbb{R}$ is an interval, be midconvex, that is $$f\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$$ for all $x,y \in I$. Assume that for some ...
6
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3answers
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Convexity and Affineness

In reading about convex optimization, the author states that all convex sets are affine. Are affinity and convexity equivalent? If I understand, both definitions incorporate the notion that a set is ...
3
votes
0answers
391 views

Modification of Mazur's lemma

Working through Brézis and solving an exercise I have a question about my solution. It's well known that, if $x_n$ converges weakly to $ x $ in a Banach space $X$, then there exists a sequence ...
3
votes
3answers
202 views

Quantifying convexity

What methods exist to quantify convexity. Yes, a set is convex if the the line between two points in the set is contained in the set, but is there a measure of how convex a set is? If so, what is ...
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1answer
104 views

Is there a name for this lattice theoretic property?

Suppose that $\langle L, \wedge, 0 \rangle$ is a lower semilattice with least element $0$. For preliminary notation, for all $b \in L$ define ${\downarrow}b = \{ a \in L \colon a \leq b \} $. Here ...
3
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1answer
147 views

Convexity of a set in Hilbert space

Let $H$ be a Hilbert space and $\left\{ e_{i}\right\} _{i=1}^{\infty}$ an orthonormal system. I need to prove that the following set is a convex set: $$C=\left\{ x\in ...
2
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3answers
135 views

Problem relating convex sets and optimization

I am working on a microeconomics problem, but I have just kind of just boiled down to the following problem involving convex sets. I have a convex set of vectors in $\mathbb{R^n_+}$ of the form ...
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1answer
73 views

Can i get every face of a polytope by taking a facet (of a facet (of a facet (…))) of the polytope?

Let $P$ be a polytope, i.e. a convex subset of a finite-dimensional real vector space with finitely many extreme points. Let $F$ be a proper face of the polytope, i.e. a subset $F \subset P$ such that ...
4
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1answer
728 views

Proving Helly's theorem

The problem is to prove Helly's theorem for the case, when the convex bodies are polytopes, by using linear programming duality. Helly's theorem Let $B_{1},...,B_{m}$ be a collection of convex ...
4
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1answer
77 views

A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
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2answers
474 views

Two definitions of a face of a convex set: are they equivalent?

I am used to the following definition of a (proper) face of a polytope: A nonempty convex subset $F$ of a polytope $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and ...
8
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1answer
894 views

A real differentiable function is convex if and only if its derivative is monotonically increasing

I'm working on a problem in baby Rudin, Chapter 5 Exercise 14 reads: Let $f$ be a differentiable real function defined in $(a,b)$. Prove that $f$ is convex if and only if $f'$ is ...
3
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0answers
417 views

a function and its epigraph

The epigraph of a function $f:\mathbb{R}^{n}\to [-\infty,+\infty]$ is the set of points $(x,\mu)\in\mathbb{R}^{n+1}$ satisfying $f(x)\leq \mu$, and is denoted by $\mathrm{epi}(f)$. I somehow got ...