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.

learn more… | top users | synonyms (3)

4
votes
3answers
147 views

Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
1
vote
1answer
16 views

Proving convexity from 2-dimensional convexity

I have a function $f(x_1,x_2,\ldots,x_m):\mathbb{R}^m\rightarrow \mathbb{R}$ ($m\geq 2$) that is jointly convex in $x_i$ and $x_j$ for all $i$ and $j$. Can I prove that this function is convex in ...
0
votes
0answers
34 views

How to deal with an $xy\le 1$ constraint?

I have to solve the following optimization problem: $$ \begin{align*} \min_{x,y} &\{-x-y\} \\ \text{such that} \\ y &\ge 3 \\ y &\le 30 \\ x &\ge 0 \\ xy &\le 1 \\ \end{align*} $$ ...
1
vote
1answer
25 views

A bound (dominated function) for $\cosh^2\left(t\sqrt{1-\gamma^2}\right)$

I would like to bound $$\cosh^2\left(t\sqrt{1-\gamma^2}\right)$$ for all $t\in\mathbb{R}$, where $\gamma^2\leq1$. How can I do such thing? This inequality maybe useful cosh x inequality
0
votes
0answers
24 views

Showing that $f$ is convex given that $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), \;\;\forall x,y \in \mathbb{R}^n$

Assume that $f$ is continuously differentiable and that for some constant $c > 0,$ the gradient $\nabla f$ satisfies, \begin{equation} (\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), ...
0
votes
0answers
25 views

How to prove that Bezier(t) polynomial lies in convex hull of points (i/n,ai) for i from 1 to n

I think i should prove firstly that: Bn,$x(t)$ for t between $0$ and $1$ lies inside the convex hull of the points $(k/n, xk)$. I know only that$ k/n$ = max between $0$ and $1$ and i found that Bezier ...
1
vote
0answers
29 views

How to say $\text {log}\ \ a^{-1} \geq 1-a$ from the concavity of $\text{log}(\cdot)$

I am reading a paper and confront the following small trick: $\text {log}\ \ a^{-1} \geq 1-a$, where $0\leq a \leq1$. By the concavity of $\text{log}(\cdot)$. From the formula: ...
0
votes
1answer
20 views

Convexify $x\le a+by^2$

I have the following non-convex constraint: $$ x\le a+by^2\quad\text{where}\quad a,b>0,\,y\in[0,y_{max}]\text{ and }a\approx by_{max}^2 $$ On a drawing, it looks something like this: The above ...
0
votes
1answer
17 views

How to prove that the right derivative of a convex function is right continuous?

let $f$ be a convex function and $D^+f$ be the right-derivative of $f$, I want to show that $D^+f$ is right continuous. first I want to use the $\epsilon-\delta$: by the definition of $D^+f$, ...
0
votes
0answers
4 views

(M,N) J-convex functions

During my analysis course, our teacher told us about (M,N) J-convex functions and quasi-arithmetic means. Do you know any article I could find out more information? Thank you!
0
votes
1answer
58 views

How to show $\partial A = \varnothing \Rightarrow A=R^n$ [closed]

Let $A\subset R^n$ and dim$A=n$, $\partial A$ is the relative boundary of $A$. If $\partial A=\varnothing$ how to show $A$ is $R^n$ ? Picture below is from XX page of Schneider R.-Convex Bodies_ ...
0
votes
1answer
24 views

Integral as a member of the closure of the convex hull of the integrand

Suppose that $X$ is compact and metric and let $g:X\to\mathbb R$ be a Borel map. Let $\mu$ be a Borel probability measure on $X$. Then it seems that $\int_Xgd\mu$ is a member of the closure of the ...
0
votes
0answers
11 views

reference for convex function results

Here is a simple property of a concave function from $\mathbb{R}$ to $\mathbb{R}$, Given $x,x'\in \mathbb{R}$ with $x'>x$, if $\exists \kappa'\in (0,1)$ such that ...
0
votes
1answer
26 views

Let B be set of all twice differentialbe function $ f(0)=1, f'(0)=-1$ . .. Find supremum of $ {(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $ f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
1
vote
1answer
15 views

Convexity of a function over a vectorial space

Consider $\mathcal{V}$ the set of vectors $X$ whose values $x_i$ are all positive. Then, consider the function f : $\mathcal{V} \rightarrow \mathbb{R} ; > ...
1
vote
0answers
41 views

Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$. Consider the function $h_K:\mathbb{R}^d\rightarrow ...
3
votes
4answers
62 views

Show convexity of a function via inequalities

I am stuck with deriving the convexity of the function $$ f(x) = \sqrt{1 + x^2} $$ from first principles, that is I would like to show that for any $x,y \in \mathbb R$ and $\lambda \in (0,1)$ we ...
0
votes
0answers
7 views

Building a convex set out of two convex sets where each extremal point of one set shares and edge with each extremal point of the other [duplicate]

Consider a convex set $P$ with two faces $f_1, f_2$ s.t. all extreme points of the convex set belong to either $f_1$ or $f_2$ (but none blong to both - the two faces are disjoint in the set of ...
0
votes
0answers
15 views

How do I prove this simple result for the face structure of convex sets?

I have a convex set $P$ with faces $f_1, f_2$ such that all extremal points of the convex set belong to either $f_1$ or $f_2$ (the faces are disjoint and cover $P$). How can I prove that if every ...
1
vote
1answer
21 views

Convexity of multi variate functions

Let $f:\mathbb{R}^m\rightarrow \mathbb{R}$ be a smooth function. I know $f(x)$ is convex if its Hessian ($\frac{\partial^2 f(x)}{\partial x\partial x^T}$) is positive semi-definite. Now, let ...
2
votes
0answers
18 views

Is $(\sum_{i=1}^n a_i{x_i}^2)(\sum_{i=1}^n b_i{x_i}^2)$ convex?

I was trying to solve a problem and at a point I needed that $$(\sum_{i=1}^n a_i{x_i}^2)(\sum_{i=1}^n b_i{x_i}^2),$$ $a_i,b_i >0$ is convex. So, I tried instead to prove that $x^2y^2$ is convex( ...
2
votes
2answers
28 views

How can I prove the concavity of $f(p_1,p_2,\ldots,p_n) = \sum_{i = 1}^n p_i(1-p_i)$?

Assume $p_n$ is the probability of being in class $n$ which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$ I need to come up with a concave function that show the relation between $p_1$ and ...
3
votes
2answers
76 views

Can you prove that this function is convex? $\sqrt{2x_1^2+3x_2^2+x_3^2+4x_1x_2+7} + (x_1^2+x_2^2+x_3^2+1)^2$.

My analysis: The second term can be proven to be convex as follows. It is basically a composition of norm with an affine transformation to the power of four: $(x_1^2+x_2^2+x_3^2+1)^2 = \|(x_1^2, ...
0
votes
0answers
5 views

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x$?

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x=[-115/588, -95/588, 5/14]^T$? Pseudoconvexity: If $\nabla f(\bar x)^T(x-\bar x)\ge0$, then $f(x)\ge f(\bar x)$ for any $x\in \mathbb{R^3}$ (in this ...
1
vote
1answer
37 views

$xy \leq \frac{x^p}{p}+\frac{y^q}{q}$

I struggled to show that this is true when $x,y >0$, $p>1$ and $q=\frac{p}{p-1}$. But, I managed to show that if we assume that $x \geq 1$ the assertion is possibly false only for a finite ...
0
votes
0answers
18 views

Convexity and composition of functions

A function $g(f(x))$ is convex if $g$ and $f$ are convex and $g$ is non-decreasing, what happens if $g(f_1(x),f_2(x),...,f_m(x))$ where $x = (x_1,...,x_n)$. Is $g$ convex if each $f_i$ is convex in ...
0
votes
1answer
25 views

Is every convex cone a manifold?

Let $C \subseteq \mathbb{R}^n\setminus \{0 \}$ be a connected convex cone*. Question: Is $C$ always a topological manifold (perhaps with boundary)? A smooth one? Does anything change if we do not ...
1
vote
0answers
40 views

Log-convexity of completely montone sequences

Let $s_0, s_1, \ldots$ be a completely monotone sequence. This means that, defining \begin{align*} (\nabla s)_n &= s_{n}-s_{n+1}\quad\text{and}\\ (\nabla^{r+1}s)_n &= (\nabla^{r}s)_n - ...
0
votes
0answers
11 views

Proof of $A\ne \mathrm{conv}~ \mathrm{relbd} A\Rightarrow A$ is a flat or a half-space

Picture below is from the 16 page of Schneider R. "Convex Bodies--The Brunn-Minkowski Theory". I am fuzzy with the red line, what is the translate of $H^-$ ? And what is the effects of convexity ...
0
votes
2answers
60 views

$\big\langle\nabla f(x)-\nabla f(y),\,(x-y)\big\rangle\ge m\,\left\| x-y\right\|^2,\;m>0\;$ for strictly and strong convex function [closed]

Prove that $\,\big\langle\nabla f(x) - \nabla f(y),\, (x-y)\big\rangle \geq m\,\left\lVert x-y\right\rVert^2, \;\,m > 0\,$ for strong convex function $f: \mathbb R^n \to\mathbb R$. Is it true ...
0
votes
1answer
19 views

Lipschitz implies bounded gradient

Assume $f:\mathbb{R}^n \to \mathbb{R}$ is convex, and $L$-Lipschitz, so $|f(x)-f(y)|\leq L\|x-y\|$. I would like to show that $\|\nabla f(x)\|\leq L$. In one dimension this is a straightforward ...
1
vote
0answers
24 views

Is there a way to determine if the Convex Hull of two polyhedra is going to be huge?

So in this post: Faster Algorithms for Convex Hulls I was interested in determining if a convex hull of two $n$ dimensional polyhedra can be computed quickly, and the answer was in general: no, ...
1
vote
1answer
58 views

A consequence of the convexity of $ f(x) = x \log x $

I verified that $f:\mathbb{R_{+}^{*}} \rightarrow \mathbb{R}, f(x) = x \log x $ is convex, since it is twice differentiable and $f''(x) = \frac{1}{x}$ is positive for the domain. But my teacher asked ...
3
votes
1answer
41 views

Is $\Gamma(\alpha, 0, x)$ log-concave as a function of $x$ (for fixed $\alpha$)?

The incomplete Gamma function is defined as: $$\Gamma(\alpha, 0, x) = \int_0^x t^{\alpha - 1} e^{-t} dt$$ Let $\alpha \ge 1$ be a fixed number. Is $\Gamma(\alpha, 0, x)$ log-concave, as a function ...
1
vote
0answers
83 views

Some intuition on the support function of a convex set

I have some doubts on the interpretation and properties of the support function of a convex subset of $\mathbb{R}^d$. (1)Let $K$ be a convex set in $\mathbb{R}^d$. (2) The support function of $K$ is ...
1
vote
2answers
30 views

Is a convex cone a convex polyhedron?

Say that I have a convex cone $C=\{t|Ax = t, x\geq 0\}$. where $x\in R^n$, and $t\in R^m$, $A\in R^m\times R^n$. Can I say that this is a convex polyhedron? and why? EDIT: Just in order to avoid ...
1
vote
1answer
43 views

Lagrange's theorem and convex functions

Let f:U⊂ $\mathbb{R}^n$--->$\mathbb{R}$ a $C^1$ function with U being convex and an open set. Let g: U ⊂ $\mathbb{R}^n$ ---> $\mathbb{R}^m$ (with m smaller than n) an affine application. Let M={x $ ...
0
votes
0answers
35 views

convex hull of a lower hemicontinuous correspondence is lower hemicontinuous

Let $(X, \mathcal{T})$ and $(Y, \mathcal{T}')$ be two topological spaces. We call a correspondence (set-valued function) $F: X \rightarrow 2^Y$ lower semicontinuous if for every $G \in \mathcal{T}'$ ...
0
votes
1answer
20 views

Locally convex spaces - is any space that contains a locally convex space as a subspace, also locally convex?

Given $E$, a locally convex space (l.c.s.) and $E\subseteq F$ where $F$ is another subspace of a larger vector space. The inclusion is strict since I know there exists a $y\in F\backslash E$. I have ...
0
votes
0answers
10 views

Characterization of projective convexity

Let $K$ be a closed set in projective space $\mathbb P^n$. Is it true that $K$ is "projectively convex", i.e., its intersection with every line is connected, if and only if it is the projectivization ...
0
votes
1answer
22 views

Proof of property of unique nearest point equal to convex.

In the picture below, I really don't know how to use the compactness to get there will be a ball $C$ with maximal radius. I report this book for my classmate and teacher. I tell they the $r$ has a ...
0
votes
0answers
17 views

If $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are positive, non-increasing and convex functions, then $F(x,y) = f(x)g(y)$ is quasiconvex.

Hypothesis: $\forall x_{1},x_{2}\in \mathbb{R}, \forall \lambda \in [0,1], f(\lambda x_{1} + (1- \lambda) x_{2}) \leq \lambda f(x_{1}) + (1- \lambda) f(x_{2})$ $\forall x_{1},x_{2}\in \mathbb{R}, ...
0
votes
1answer
26 views

Extreme points of unit ball of $l_1(\mathbb{N})$

Let $K$ be the closed unit ball of $l_1(\mathbb{N})$ over real numbers. Show that $$ Ext(K)= \{\pm e_n: e_n=(0,\ldots,1,0,\ldots)\}. $$ My attempt: I could prove that $\{\pm e_n: ...
0
votes
0answers
14 views

Solve convex optimization problem with objective function having power \alpha >1

I am not able to figure out how to get the explicit solution of the following minimization problem: $$\min_{\mathbf{w}\in \mathbb{R}^n} = \sum_{i = ...
1
vote
0answers
37 views

Need to prove that convex property is the intersection of an increasing and decreasing property for graph

I need to prove that any convex property for graphs can always be expressed as the intersection of an increasing property and decreasing property for graph, specifically that: $\forall A\subset ...
0
votes
0answers
20 views

general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times ...
3
votes
1answer
47 views

Convex set: extreme points and distance to the origin

I'm fairly sure the following is true, although I wouldn't mind being proven wrong. If true, I would like to see an elegant proof, as my attempts are kind of messy. Let $K\subset\mathbb R^2$ be a ...
0
votes
0answers
27 views

convexity of composite function

There is a composite function F(x)=f(g(x)): R -> R where f(v): R^2 -> R and g(x): R -> R^2. It is given that f(v) is convex on v and v = g(x) gives a map R -> R^2. If x is in a convex set, is F(x) ...
0
votes
1answer
67 views

Convex set without zero

Let $\emptyset \neq A \subset \mathbb{R}^n$ be a convex set with $0 \notin A$. Then there exist a $v \in \mathbb{R}^n$ such that $v \cdot a \geq 0$ for all $a \in A$ and there exists $a_0 \in A$ with ...
0
votes
0answers
18 views

An inequality of a multivariate function: $f(x_1,…,x_n) \geq \frac{1}{n}\sum_i^n f(x_i,x_i,…,x_i) $

Let us assume we have a non linear function $f : \Bbb R^{n+} \to \Bbb R ^+$, and let $x = \{x_1, x_2 , ..., x_n\}$, $x_i \in \Bbb{R}^+$, further define $\bar{x} = \frac{1}{n}\sum_i^n x_i$. My ...