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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Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
0
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1answer
23 views

Convex hull of $\{ \Vert x \Vert = 1 \}$ is closed in strictly convex space

I'm trying to show that the convex hull of $\{ \Vert x \Vert = 1\}$ is closed in a strictly convex Banach-space. I don't know how to tackle the problem. Are there any nice characterizations for a ...
0
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2answers
35 views

Is the logarithm of sum of multiple variables with the constraint on sum of them Concave?

I know that without any constraint $log \sum_{i=1:1:m} \alpha_i C_i $ is not Concave but I am wondering is this function Concave when we have the constraint that $ \sum_{i=1:1:m} \alpha_i =1 $ and ...
1
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0answers
19 views

Core points of a convex set

In the book of Gamelin "Unifrom Algebras" I found the following definition: Let $K \subset V$ be a convex set in a vector space. An element $x \in K$ is called core point of $K$ if whenever $z ...
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20 views

Analytical algorithm to obtain solution to convex optimization problem.

Assume a vector $\vec{P}$ with N elements $\in \mathbb{R}^+$ and constants $T_P$ and $\epsilon$. The vector $\vec{P}$ is arranged in a column $(N\times 1)$. Consider the problem: $$ \begin{aligned} ...
6
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3answers
180 views

Show that $\left(\int_{0}^{1}\sqrt{f(x)^2+g(x)^2}\ dx\right)^2 \geq \left(\int_{0}^{1} f(x)\ dx\right)^2 + \left(\int_{0}^{1} g(x)\ dx\right)^2$ [closed]

Show that $$ \left( \int_{0}^{1} \sqrt{f(x)^2+g(x)^2}\ \text{d}x \right)^2 \geq \left( \int_{0}^{1} f(x)\ \text{d}x\right)^2 + \left( \int_{0}^{1} g(x)\ \text{d}x \right)^2 $$ where $f$ and $g$ ...
0
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1answer
48 views

Functional analysis - check that a closed subspace of a Hilbert space is convex

Suppose that V is a Hilbert space over $F$ and $W$ is a closed subspace of $V$ . Then for every $x \in V$ , there exist unique $y \in W$ and $z \in$ (the orthogonal compliment of $W$) such that $x = y ...
0
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0answers
37 views

is this function an ill-shape convex function?

I have a function with parameter $\vec{{\alpha}}$ where it is formulated by the formula: $$ f(D|\alpha)=n_1{\alpha}_{1}+...+n_m \alpha_m -Nlog \sum_{i=1:m} exp(\alpha_{i}+g_i(D)) $$ where $g_i{(D)}$ ...
6
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1answer
107 views

Inf-convolution, two basic questions?

Let $E$ be a normed vector space. Given two functions $\varphi$, $\psi : E \to (-\infty, +\infty]$, one defines the inf-convolution of $\varphi$ and $\psi$ as follows: for every $x \in E$, ...
1
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2answers
42 views

Minimising a convex set. Is set of solutions convex?

We are minimising a convex function on a non-empty set defined by linear constraints (equalities and inequalities). $X^O$ is the set of all optimal solutions and we assume it is non-empty. Is it true ...
3
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0answers
17 views

Supporting hyperplane to a compact, convex set in Hilbert space at a given boundary point

Does one always exist? I see that it is true in finite dimensions.
1
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1answer
36 views

sum of two cones

A non-empty set $K$ of a vector space is called a cone if it satisfies the following: $ K +K \subseteq K,$ $\alpha K \subseteq K$ for all $\alpha \ge 0,$ $K \cap (-K) ={0}$. Let $K_{1}$ and ...
0
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0answers
21 views

All facets' coefficients contain only integers -1,0,1

Suppose a polytope $C\subset \mathbb R^{kl}$ is the $l$-product of $k-1$-simplex with extreme points containing coordinates $0$ or $1$ in each coordinate. A linear transformation is given by ...
0
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1answer
23 views

integer valued outer normal vectors

Suppose a bounded polyhedra $C$ is given by $$x\in \mathbb R^n: Ax\leq b$$ The matrix $A\in\mathbb R^{m\times n}$ contains only elements from $\{-1,0,1\}$, which implies the outer normal vectors of ...
0
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1answer
28 views

How large can the set $\partial ( \bigcup_i B_i ) \setminus \bigcup_i ( \partial B_i )$ be, where the $B_i$ are open balls in $\mathbb{R}^n$?

Suppose that $$E=\bigcup_{i=1}^{\infty}B_i,$$ where the $B_i$ are open balls in $\mathbb{R}^n$ and for $i\ne j$, $B_i \cap B_j = \emptyset$. We know that generally $$\bigcup_i \partial B_i \subsetneq ...
4
votes
1answer
132 views

Obtaining an estimate for the Lagrangian $L=H^*$ from the Hamiltonian $H$

This is from C. Evans' PDE book, page 130. The convex function $H:\mathbb{R}^n\to\mathbb{R}$ is $C^2$ and satisfies $$ H\big(\frac{p_1+p_2}{2}\big) \leq \frac{1}{2}H(p_1) + \frac{1}{2}H(p_2) - ...
6
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1answer
132 views

Derivative of intersection volume

Let $K$ be a convex body in $\mathbb{R}^n$ and set $f:\textrm{SL}(n)\rightarrow \mathbb{R}$ as $f(T)=\textrm{Vol}_n (TB\cap K)$ where $B$ is the Euclidean unit ball. How can we find extreme points of ...
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46 views

How to find the tangent cone to a set in a point?

Let $S\in R^{n}$ is a set and $x\in S$. We define tangent cone of $S$ in $x$ as: $$T_{S}(x)=\{z\in R^{n}:\exists (x_{k}), x_{k}\in S, x_{k}\rightarrow x, \exists (y_{k}), y_{k}>0, ...
3
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1answer
16 views

Regularization by inf-convolution

Let $E$ be a n.v.s. and let $\varphi: E \to (-\infty, +\infty]$ be a convex l.s.c. function such that $\varphi \not\equiv +\infty$. Let$$\varphi_n(x) = \inf_{y \in E} \{n\|x - y\| + \varphi(y)\}.$$ ...
3
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2answers
58 views

Is finding the second derivative of $\sqrt[3]{\vert x\vert}$ the best method to determine if it is convex?

I have an exercise where I have to tell on which intervals a function is concave or convex. I usually do it using second derivative, but I would like to know if there is a simpler way of doing so, ...
3
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1answer
29 views

First order condition in constrained optimization: Alternative characterization via normal cones

Consider the following constrained optimization: Min $f(x)$, $x\in C\subset R^n$, where C is convex. We know that one characterization of a local minimum (necessary condition) is the following: ...
0
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1answer
32 views

Convexity of the Riemann-zeta function without derivative

Proving that the $\zeta$ function is convex on $(1,+\infty)$ is pretty simple if we use the derivative, but is there a proof without using derivative? I'm allowed to use just the definition of the ...
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2answers
28 views

preservation of extreme points under linear transformation

Suppose $\{e_1,...,e_N\}$ is the set of all extreme points of a compact convex subset $X\subset\mathbb R^n$. $L: \mathbb R^n\to \mathbb R^m$ is a linear transformation. $L$ is surjective but is not ...
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0answers
49 views

How to reshape a nonlinear inquality into a linear matrix inequality?

We have these two nonlinear inequalities (I): $$x^2+y^2>0$$ $$3x^2+3y^2-4y^6>0$$ We want to represent this problem as a Linear Matrix Inequality Problem, i.e, we want to derive a positive ...
2
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2answers
56 views

Convexity implies absolute continuity?

The following is taken from an exam: $f:[a,b]\rightarrow\mathbb{R}$ is convex implies $f$ is absolutely continuous (recall $f'$ exists a.e.) One has local Lipschitz-ness by convexity, but how to ...
0
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2answers
17 views

Convex Set with Empty Interior Lies in an Affine Set

In Section 2.5.2 of the book Convex Optimization by Boyd and Vandenberghe, the authors mentioned without proving that "a convex set in $\mathbb{R}^n$ with empty interior must lie in an affine set of ...
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143 views

Is $\frac{1}{2}\|x\|^2$ the only function that is equal to its convex conjugate?

Is $\frac{1}{2}\|x\|^2$ the only function that is equal to its convex conjugate? The convex conjugate is defined as $$ f^{*}(x) = \sup_y\{\langle x, y\rangle - f(y)\}. $$
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1answer
32 views

Tangent lines for convex functions

In theorem 1 here, the author says that if $\phi$ is a convex function on $(a,b)$ then for every point $c\in (a,b)$ there exists a line $L$ that passes through $c$ such that the graph of $\phi$ lies ...
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2answers
74 views

Balanced cutting of a convex polygon

Given a convex polygon $C$ and a number $R\geq 1$, say that a point $x$ is an $R$-balance-point of $C$ if every line through $x$ divides $C$ to two parts $C_1,C_2$ such that: $$1/R \leq ...
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0answers
36 views

Is $A$ convex if and only if $-\ln(i_{A})$ is convex?

Is it correct that we have : $A \subset \mathbb{R}^n$ is convex if and only if $-\ln(i_{A})$ is a convex function? where here $i_A$ takes the value 1 at $x \in A$ and $+\infty$ elsewhere. I have ...
0
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1answer
52 views

an example of a non convex ideal [closed]

As an example of a non convex ideal we have in Gillman and Jerison, Rings of Continuous Functions, 1976, Exercise 5E(1), the ideal $I= (|\operatorname{id}_{\mathbb R}|)$ in $C(\mathbb R)$. I need to ...
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32 views

translate of a homothet of a convex body

Suppose we have given a convex body $K \subset \mathbb{R}^2$. How can we prove that it contains a translate of its homothet $-\frac{1}{2} K$? hint: take three vertices $A, B$ and $C$ of the convex ...
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24 views

Proving concavity of a function

Let $f:[a,b]\rightarrow\mathbb{R}$ be twice-differentiable. Show that $f$ is concave if and only if $f''(x)\leq0$ for all $x\in[a,b]$. Moreover, if $f''<0$ for all $x\in[a,b]$, $f$ is strictly ...
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37 views

Eigenvalues of a convex sum of two positive-definite matrices (with a lot of collateral information about them)

I have two positive-definite matrices written as a sum of rank-1 matrices $P_i$ (not projectors) $$ S_1=P_1+P_2+P_3+P_4,\\ S_2=P_5+P_6+P_7+P_8. $$ It is not an eigendecomposition ($P_i$s are not ...
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2answers
34 views

Convexity and local maxima

If a continuous function $f$ on $(a,b)$ is not convex, there is some choice of number $m$ so that $g(x)=f(x)+mx$ has a local maximum at a point $z$ inside the interval $(a,b)$. Is that true?
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29 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
4
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1answer
56 views

Is it sufficient for convexity?

Is it true that the sufficient and necessary condition for a real-valued function $f$ to be convex on an open interval $I$ is that i) $f$ is continuous on $I$ and ii) $\bar{D}_2 f \geq 0$, where $$ ...
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2answers
51 views

Convexity of exponential function [closed]

How to prove that the convexity of exponential function? It is not allowed to use second derivative of $e^x$.
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46 views

How can we find the area of the triangle which covers a finite point set in $\mathbb{R}^2$ by using the interior triangles with specified area?

Suppose we have given a finite point set $X \subset \mathbb{R}^2$ in a way that any triangle made by vertices of $X$ has area at most 1. How can we prove that there is a triangle of area 4 which is ...
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2answers
41 views

do infinite family of lines in $\mathbb{R}^2$ have a common point by knowing that any three of them have common point?

Suppose we have given an infinite family of lines; say $\mathfrak{F}$, in the plane $\mathbb{R}^2$ such that any three of the lines in $\mathfrak{F}$ have a common point. How can we prove that all ...
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2answers
36 views

A subset $K$ of $L^1$ such that is convex, absorbent and balanced, but not neighborhood of $0$.

It is well-known that, if $(\Omega,\mathcal{F},P)$ is an atomless probability space, then $L^1$ is barreled, in the sense that every subset $K$ which is closed, convex, absorbent and balanced is a ...
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1answer
37 views

Proof of convexity of in a quadratic function

Let $ X= \{(x_1,d_1),(x_2,d_2),...,(x_n,d_n)\}$ where $x_i$ for $i=1,...,n$ are variable and $d_i$ for $i=1,...,n$ have constant values, then we define: $$ F(X) = \min\sum_{i=1}^{n} ...
3
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33 views

open convex cone coincides with the interior of its closure

Let $V$ be a finite dimensional real Euclidean space and $C$ be an open convex cone in $V$. I need to prove that $C = int(\overline{C})$. I proved that $C \subseteq int(\overline{C})$. I have ...
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30 views

On the (strong) convexity of a function. Why does it stop to be strongly convex?

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ be given as follows $$ f(\mathbf{w})=\frac{\lambda}{2}\lVert\mathbf{w}\rVert^2+\frac{1}{k}\sum_{i=1}^{k}\mathcal{L}(\mathbf{w};\mathbf{x}_i), $$ where the so-called ...
0
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1answer
35 views

Properly or strictly separated sets

Let $A=\{ x,y,z: x,y,z\in[0,1] \}$ and $B=\{(x-2)^{2}+(y-2)^{2}+(z-2)^{2}\le 1\}$. Show if the sets $A$ and $B$ can be properly or strictly separated. Does anyone know the solution of this problem?
4
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73 views

The reverse pizza problem .

The pizza problem is a fairly well-known problem which sounds like this : You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . ...
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36 views

how to find the maximum area of a two rectangle under a parabola

Starting from a very basic concept, what is the largest triangle to be drawn under the function $f(x)$ as shown in the figure. Picking an arbitrary point on the x-axis $(x, 0)$ and its mirror $(-x, ...
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1answer
9 views

Separition arguments and support functions

Show that if $F,G \subseteq E$ are compact convex sets such that $\sigma_F=\sigma_G$ then $F=G$ (this requires a separation argument) where$$\sigma _F (x) := \max\{\langle x, u\rangle : u ∈ F\}.$$ ...
5
votes
1answer
38 views

How to show that $y^T x - \frac{1}{2}x^T Q x$ is bounded above?

Strictly convex quadric function. Consider $f(x)=\frac{1}{2}x^TQx$, With $Q\in S_{++}^n$. The function $y^T x - \frac{1}{2}x^T Q x$ is bounded above as a function of $x$ for all $y$. It attaints its ...
2
votes
1answer
19 views

When is a functional a convex combination of other functionals?

Suppose that $f, g_1,...,g_n$ are functionals defined on a normed vector space $E$ and that for each $x \in E$ we have that $f(x)$ is in the convex hull of $\{g_1(x),...,g_n(x)\}$. Does this imply ...