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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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 ...
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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.
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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 ...
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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 ...
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24 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 ...
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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 ...
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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) - ...
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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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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, ...
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21 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$$\varphi_n(x) = \inf_{y \in E} \{n\|x - y\| + \varphi(y)\}.$$ ...
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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, ...
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31 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: ...
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1answer
39 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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36 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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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 ...
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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 ...
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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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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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36 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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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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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 ...
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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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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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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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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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35 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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36 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 / ...
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60 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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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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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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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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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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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} ...
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open convex cone coincides with the interior of its closure [duplicate]

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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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 ...
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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?
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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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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
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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$$\sigma _F (x) := \max\{\langle x, u\rangle : u ∈ F\}.$$ ...
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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 ...
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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 ...
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Support function and convexity

Let $A, B, C$ be compact convex sets in $\Bbb R^n$ such that $A + C = B + C$. The purpose of this problem is to prove that $A = B$. Define the support function $$\sigma _A (x) := \max\{\langle x, ...
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Show that the following is a convex set

I've been banging my head against the wall trying to handle these proofs for two hours now, it seems very simple but I guess I need a hand starting out. I hope I at least know what to show: Show that ...
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How to use Farkas' lemma?

How can I prove, that the set $$P = \{(x, y) \in \mathbb{R}^{n+m} : Ax + By \geq c, \: x \geq 0^n, \: y\geq 0^m \}, $$ where $B \in \mathbb{R}^{m \times m} \;$ is positive semidefinite matrix, $A ...
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boundedness of convex functions

Let $X$ be a vector space, $\Omega$ a convex subset thereof and $f:\Omega \to \mathbb R$ a convex function. Then $f$ need not be bounded from below - not even if it is strictly convex, as the example ...
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Equivalent characterization of quasi-concavity [duplicate]

For $f: \mathbb R^n \to \mathbb R$ prove that the two statements are equal: For all $x,y \in \mathbb R^n$ and for all $t \in [0,1]$, $f(tx+(1−t)y)\geq \min (f(x),f(y))$ For all $k \in \mathbb R$, ...
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Chek the convexity of a set via affine functions

The set: $\{x| x^TPx \leq (c^Tx)^2, c^Tx \geq 0 \}$ where $P \in S^n_+$(SPD matrices) and $c\in R^n$, is convex, since it is the inverse image of the second-order cone, $\{ (z,t) | z^Tz \leq t^2, ...
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Differentiability of convex functions except at countably many points

There is this result in Notions of Convexity, Hormander. The relevant part of it reads: let $f$ be convex in an interval $I$ and $x$ be an interior point. Let $f_l'$ and $f_r'$ denote left derivative ...
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Permutation and Combinatorics Problem

A function $G$ is defined on a set $S$ with size $k$ : $G(a_1,a_2,a_3,\ldots,a_k)$. $G(a_1,a_2,a_3,\ldots,a_k) = 1$ if and only if a convex polygon can be created by taking these $k$ elements as the ...
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Is this combination of convex functional is still convex?

Let $u$, $v\in C_c^\infty$ and $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. We also assume that $0\leq v\leq 1$. Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx. $$ Do we have ...