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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Showing that $f$ is convex given that $f(\frac{x+y}2)\le\frac{f(x)+f(y)}2$

Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that $f$ is convex. Thanks. Edit: I'm sorry for all the confusion. $f$ is assumed to be continuous on an ...
14
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4answers
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Why does a convex set have the same interior points as its closure?

Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just ...
12
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1answer
779 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 ...
13
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2answers
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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.
4
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1answer
172 views

Convexity of $\mathrm{trace}(S) + m^2\mathrm{trace}(S^{-2})$

I have the following function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ where $S\in \mathcal{M}_{m,m}$ symmetric positive definite matrix. I'm trying to prove the convexity of this function ...
2
votes
1answer
77 views

Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
16
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6answers
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can any continuous function be represented as a sum of convex and concave function?

I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave. There could be ...
7
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1answer
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Is the trace of inverse matrix convex?

Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex. Actually I know that the trace of a symmetric positive definite ...
11
votes
2answers
283 views

Convexity of a complicated function

Let $\mathbb{S}$ be a 2-D convex set in the positive quadrant. Let us define \begin{align} y_L=\min_{(x,y)\in\mathbb{S} }y \\ y_R=\max_{(x,y)\in\mathbb{S} }y \end{align} For any positive number ...
5
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1answer
110 views

Equations for interior of Platonic solids

It is well-known that for Platonic solids: The interior of cube a.k.a. hexahedron can be described with inequality $\max\{|x|,|y|,|z|\}<a$. The interior of octahedron is $|x|+|y|+|z|<a$. But ...
5
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1answer
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Continuity of a convex function

I'm trying to solve the following problem: Let $f:K\rightarrow \mathbb{R} $, $f$ convex and $K \subseteq \mathbb{R}^n$ convex. Then $f$ is continuous on $K$. I have proved the only case $n=1$, but ...
6
votes
1answer
867 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
2
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1answer
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$f$ is convex function iff Hessian matrix is nonnegative-definite.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le ...
12
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2answers
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Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
10
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1answer
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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 ...
11
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2answers
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How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
7
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3answers
142 views

If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then how do we ...
7
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1answer
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Convexity of the product of two functions in higher dimensions

Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both ...
4
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1answer
412 views

Limit of derivatives of convex functions

Let $(f_n)_ {n\in\mathbb{N}}$ be a sequence of convex differentiable functions on $\mathbb{R}$. Suppose that $f_n(x)\xrightarrow[n\to\infty]{}f(x)$ for all $x\in\mathbb{R}$. Let ...
1
vote
1answer
327 views

interior points and convexity

Question: Let If $P\subseteq \mathbb{R}^n$ be a convex set. show that $int(P)$ is a convex set. I know that a point $x$ is said to be the interior point of the set $P$ if there is an open ball ...
3
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1answer
122 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
3
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1answer
792 views

Prove that the convex hull of a set is the smallest convex set containing that set

How do you prove that the convex hull of A is the smallest convex set containing A? edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called ...
1
vote
2answers
263 views

Is every convex function on an open interval continuous?

Let $f:(a,b)\rightarrow \mathbb{R}$. $f$ satisfied the following property: If $\forall x_{1},x_{0},x_{2}\in(a,b)$ and $x_{1}<x_{0}<x_{2};$then$\frac{f(x_{0})-f(x_{1})}{x_{0}-x_{1}}\geq ...
6
votes
1answer
386 views

A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, ...
17
votes
1answer
587 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...
15
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1answer
317 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
5
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1answer
137 views

A convex function that is bounded on a neighborhood is Lipschitz

Let consider a normed vector space $V$. I want to prove that If $f:V\to \mathbb R$ is a convex function and if for some $x_0 \in V$ the function is bounded on a neighborhood $W$ of $x_0$, then ...
4
votes
1answer
518 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
2answers
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Second derivative positive $\implies$ convex

In proof of the following theorem; If $f$ has a second derivative that is non-negative (positive) over an interval then $f$ is convex (strictly convex). $f$ is in real number space., the book I ...
2
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1answer
39 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
2
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2answers
326 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
2
votes
2answers
609 views

Convex combination in compact convex sets.

Let's $K\subset\mathbb{R}^n$ compact. If $K$ is convex then who to prove that any point of $K$ is convex combination of one or two extremal points of $K$? Intuitively, for any closed ball that is ...
0
votes
1answer
60 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...
0
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1answer
70 views

How to prove that $f$ is convex function

Let $f:(a,b) \mapsto \mathbb{R}$ be continuous function such that $f(\frac{x+y}{2})\leq \frac{1}{2}f(x) + \frac{1}{2}f(y) \;\; \forall x,y \in (a,b)$ Show that $f$ is convex function. Please give ...
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votes
3answers
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How to prove that $e^x$ is convex? [closed]

I need a help with proving convexity of $e^x$ function. We know $e^x$ is a convex function. $$f(c(x')+(1-c)(x'')) ≤ c\cdot f(x')+(1-c)\cdot f(x''), \quad 0 ≤ c ≤ 1$$ Thanks.
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votes
4answers
462 views

There is a ray from each point of unbounded convex set that is inside the set. [closed]

Let $A$ be a non-empty convex, unbounded set in $\mathbb R^n$. Prove that for each point $a \in A$, there is a non-zero vector $h \in \mathbb R^n$ such that $l = \{x \in \mathbb R^n \mid x=a+th,\ ...
10
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1answer
491 views

Surface area of a convex set less than that of its enclosing sphere?

Is the boundary measure ("surface area") of a convex set in $\mathbb{R}^n$ less than the boundary measure of it's enclosing hypersphere (smallest hypersphere that contains the set)? In 2D I've found ...
5
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1answer
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The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
3
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1answer
361 views

Continuous boundary of a convex set

Is the boundary of a compact convex set in Rn continuous? Seems like the answer should obviously be yes, but I cannot find any such result in the literature. Can somebody provide a reference (or a ...
2
votes
1answer
41 views

convexity proof of a function including ln and sums

$$f(x_1,\dots,x_n)=\sum\limits_{i=1}^nx_i\ln x_i-\left(\sum\limits_{i=1}^nx_i\right)\ln\left(\sum\limits_{i=1}^nx_i\right)\rightarrow R_{++}^n$$ How can I prove this is convex on $R_{++}^n$? I tried ...
2
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3answers
282 views

I don't understand this proof of the AM-GM inequality?

The proof uses this lemma which I understand: $\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ ...
2
votes
2answers
152 views

When does it make sense to define a generator of a set system?

In a set system, such as a topology, sigma algebra or convexity structure, a generator is defined to be a class of subsets such that the given set system is the coarsest such set system ...
2
votes
2answers
256 views

Is this set compact?

Given two densities $f$ and $g$ the squared Hellinger distance between them is defined as follows $$ S(g,f)=H^2(g,f)=\frac{1}{2}\int_{\mathbb{R}}\left(\sqrt{g(y)}-\sqrt{f(y)}\right)^2 \mbox{d}y $$ I ...
2
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1answer
576 views

Is a convex function defined on a convex open subset of $\mathbb R^n$ continuous?

Let $K$ be a convex open set in $\mathbb R^n$ and $f$ a convex function defined on $K$; how to show that $f$ is continuous?
1
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1answer
51 views

Does having positive second derivative at a point imply convexity in some neighborhood?

Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$. Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is ...
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0answers
100 views

Showing convexity of a function with the restriction over an arbitrary line

Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set. $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ ...
1
vote
2answers
162 views

On the decomposition of stochastic matrices as convex combinations of zero-one matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
1
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0answers
37 views

can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
1
vote
1answer
161 views

A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
1
vote
0answers
185 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...