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 every such function convex or concave?

I was pondering convex functions today, and the following questions naturally posed themselves. Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ $2$-limited iff for all $m,c \in \mathbb{R},$ ...
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2answers
48 views

Convexity of functions

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?) If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I ...
28
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2answers
874 views

Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any ...
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1answer
101 views

Quasi-Concavity and Quasi-Convexity

My book states that: $f$ is a quasiconcave function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: $f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ $f$ is a quasiconvex function on $U$ ...
0
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1answer
50 views

Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: ...
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1answer
34 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
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1answer
48 views

Strictly Convex and Differentiable Implies

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$? This is true if $f$ is smooth but am ...
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1answer
150 views

Gradient descent (with line search) for convex functions viewed as alternation

I have fundamental confusion about gradient descent (with line search) and the reason it works. I try to explain my view here, and please tell me where it goes wrong. Let $f: \mathbb{R}^n \to ...
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0answers
253 views

separating hyperplane theorem proof

I need help understanding the last part of this proof, the lemma they are refering too is just a lemma about convex sets that shows us that p is unique: I do not see how H* separates C and z. The ...
0
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1answer
65 views

$F(x) = f(x) + g(x) + h(x)$, where h(x) is strongly convex , is also strongly convex

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ $\newcommand{\Tr}{\operatorname{Tr}}$ Suppose $g: \mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous convex ...
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2answers
42 views

Prove that the set $C = \{x \in\Bbb R : ax\le b\}$ is convex

Prove that if a and b are real numbers, then the set $C = \{x \in\Bbb R : ax\le b\}$ is a convex set. My solution so far: To show that a set $C$ is convex it needs to be shown that for for every ...
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1answer
46 views

Convexity on a direction

In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have: A function $f$ defined on a triangle $T$ is said to be convex in the ...
2
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1answer
116 views

Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
0
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1answer
19 views

What is a $0$-sublevel set?

I read the notes of S. Boyd, and am confused about the following: $f_0(x)$ is quasiconvex. I am confused about the latter one particularly. What does it mean? Thanks!
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0answers
24 views

Show P is linear if it is convex and positive homogenious functional

it might be too simple but I couldnt show the second part L linear real space $P:L\rightarrow \Bbb R$ is called positive homogenious functional if for every $x\in L$ and $\alpha\ge 0$ , $P(\alpha ...
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1answer
22 views

Alternative characterization of a finite dimensional affine set

As the definition in the S. Boyd's textbook: My question is the following representation: What is the relationship between this representation and the definition above it? EX: sum of elements ...
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0answers
151 views

(Strictly) concave/convex function: Increasing / decreasing slope triangle?

I have the following (possibly quick) question. In a paper I am working with, the following conclusions are drawn which I have a hard time to understand. Since they are given without proof, I assume ...
2
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0answers
33 views

Extending the notion of convex hull in $\Bbb R^n$

The super convex hull of a set $A \subseteq \Bbb R^n$, is the set of all $\sum_{i=1}^{\infty}\lambda_i x_i$ such that $\lambda_i \geq 0$ and $\sum_{i=1}^{\infty}\lambda_i =1$, which is denoted by ...
2
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1answer
48 views

Find $C\subset \mathbb{R}^2$ convex unbounded such that $\vert C \vert $ is not convex?

The question is almost posted in the title and one thing to put is that $$\vert C \vert : = \big\{ ( \vert x \vert , \vert y \vert )\in [0, +\infty)^2 \, : \,\, (x, y)\in C \,\, \big\} $$ If $C$ ...
0
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2answers
130 views

Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
2
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0answers
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Krein-Milman theorem

Let $E$ be a Riesz space with order unit $u$. With $$\left\|f\right\|_u: = \inf\{\lambda \in [0,\infty): -\lambda u\leq f\leq \lambda u\} $$ $E$ becomes a normed space. The following sets are subsets ...
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1answer
28 views

Representation of half-space

For any non-zero $\mathbf{y}\in\mathbb{R}^\mathcal{l}$ one half-space through origin is defined by $H_{\mathbf{y}}^{\leq}(\mathbf{0})=\left\{\mathbf{x}\in\mathbb{R}^{\mathcal{l}}:\mathbf{y}\cdot ...
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Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
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1answer
41 views

Explain a linear function property?

Let $C(x)$ a linear function on $\mathbb{R}$. Then we have: $$ \begin{align*} C\left(S_0^1\right)&=C\left( \frac{y_m-r}{y_m-y_1}S_0^1+\frac{r-y_1}{y_m-y_1}S_0^1 \right) \\ ...
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1answer
106 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
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1answer
26 views

taking convex hull does not add extreme points

can anybody help me please? Is there a good way to prove that given a set of points, say $S = \{x_1, x_2, ..., x_n\}$, then show that the convex hull of $S$, that is, $conv(S)$ contains all the ...
3
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1answer
108 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
0
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1answer
81 views

Difference between subspace and subset

Can you give the definition of subspace and subset of $\mathbb{R}^n$ and how can I determine their dimension?
2
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1answer
69 views

Discrete Cauchy integral formula : The interior values are always convex combinations of exterior values for harmonic functions?

Let $T={\mathbb Z}^2$. For $t=(x,y)\in T$, the neighborhood $N(t)$ of $t$ is the four-point set $\lbrace x\pm 1;y\pm 1\rbrace$. A map $f:T \to {\mathbb R}$ is harmonic iff $4f(t)=\sum_{s\in ...
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How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
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1answer
101 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
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1answer
32 views

3D Convex hull in 3D Convex hull

I have two convex hull, How to check if the smaller one is wholly, partially or not inside the Bigger Convex hull?
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75 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
0
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1answer
53 views

Strong convexity of a function with cases

Given a set $S = \{x_1,\dotsc,x_n\} \subset \mathbb{R}$, is the function \begin{align} f&: (0,\infty) \to \mathbb{R} \\ f&(p) = 2p^2 + \frac{1}{n}\sum_{i=1}^n \max(0, -p^2-x_i) \end{align} ...
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1answer
47 views

Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
0
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1answer
43 views

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 - ...
0
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1answer
32 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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1answer
76 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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1answer
69 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 ...
0
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1answer
25 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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139 views

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

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 $A^*=\{x\text{ in }\Bbb R^2:ax\leqslant 1\text{ for all }a\text{ in }A\}$. I study from ...
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1answer
45 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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1answer
58 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
41 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
35 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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2answers
104 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}}- ...
4
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1answer
95 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
35 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
58 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 ...
6
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1answer
99 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: ...