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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Generalization of log-convexity (log-concavity): log-log-convexity (log-log-concavity)?

$\underline{\mathrm{Background\; on\; function\; Convexity}}$ A function, $f$, is convex if: $$f( x\theta+y(1-\theta) ) \leq \theta f(x) + (1-\theta)f(y).$$ $f$ is concave if $-f$ is convex, [1]. If ...
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1answer
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Difference quotients are increasing for $f$ convex

Problem 11A.9(c) in Spivak's Calculus (4th edition) asks the following (I'm paraphrasing): Suppose $f$ is convex. Show that $f'(a)$ exists iff $f_+'(x)$ is continuous at $a$. ($f_+'(x)$ is the ...
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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
192 views

Prove that a Hilbert space is convex of power type $2$

Let $X$ be a Banach space. For $\epsilon \in (0,2]$, define: $$\delta_X(\epsilon) = \inf_{x,y \in X}\{1 - \|\frac{1}{2}(x + y)\| : \|x\| = \|y\| = 1, \|x-y\| \ge \epsilon\}.$$ Then we say that $X$ ...
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116 views

Are strictly convex functions with positive second derivatives on compact domains strongly convex?

Claim: Let $\chi$ be a compact set. If $f''(x)>0$ for all $x\in\chi$, then $f$ is strongly convex. This seems to be true, intuitively, as I can't think of a counterexample. All of the examples ...
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40 views

On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?

Let $X$ be a Banach space. Let $p > 1$ and, consider the functional $X \to \mathbb{C}$ given by: $$x \mapsto \frac{1}{p}\|x\|^p$$ I would like the know if the above functional is convex. That ...
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51 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
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How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...
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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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172 views

Knuth's Sandwich Theorem: requesting proof clarification

The question is about F6 of Section 8 ("Elementary facts about cones") in Donald Knuth's Sandwich Theorem (http://arxiv.org/pdf/math/9312214.pdf). He claims to prove $(A \cap B)^* = A^* + B^*$ when ...
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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 ...
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142 views

Strictly Convex Functions

I am trying to show the equivalence of two definitions of strictly convex functions. Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth function. The function $f$ is strictly convex if for each ...
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Convex Sets in Functional Analysis?

Why did Bourbaki choose to study convex sets, convex functions and locally convex sets as part of the theory of topological vector spaces, and what is so important about these concepts? I'd like to ...
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1answer
216 views

How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
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2answers
60 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 ...
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287 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 $U$ ...
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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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175 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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46 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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63 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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88 views

Lipschitz in $\mathbb R^1$ implies Lipschitz along any line in $\mathbb R^k$ (for convex functions)

I'm trying to solve a homework problem (baby rudin course) and realized that Lipschitz continuity really makes the problem easier for me. However, since it is not defined in Rudin I have to prove the ...
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616 views

How to compute the image of a polyhedron under a linear transformation

Suppose $P$ is a polyhedron whose representation as a system of linear inequalities is given to us: $$ P = \{ x ~|~ Ax \leq b\}$$ Define $P'$ be the image of $P$ under the linear transformation which ...
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631 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 ...
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353 views

Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
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1answer
100 views

Convex polygon is connected by broken lines with each chain parallel to one of two vectors

Given is convex polygon $P$ on the plane. Settle if there are two vectors $\vec{a}$, $\vec{b}$ such that any two points belonging to the polygon $P$ you can connect a broken line contained in the ...
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On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
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1answer
63 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 ...
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168 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 $n+1$ ...
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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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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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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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274 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 ...
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181 views

Strictly decreasing function with a horizontal asymptote is convex?

Suppose $f$ is a strictly decreasing function with a horizontal asymptote at $t \rightarrow + \infty$. Hence, there exists a $t_{0}$ such that $\forall t>t_{0}, ~f(t)$ is a convex fuction. Is this ...
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If both $f$ and $-f$ are convex functions, then $f$ is affine

Prove that if both $f$ and $-f$ are convex functions, then $f$ is affine My attempt If $f$ is convex, $f(\lambda y +(1-\lambda)x) \le \lambda f(y) + (1-\lambda)f(x)$ If $-f$ is convex, ...
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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 $C$ ...
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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?
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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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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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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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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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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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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 ...
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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 ...
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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?
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82 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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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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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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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 ...
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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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59 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} ...