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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finding a counter example to Caratheodory's convex hull theorem for infinite dimentional space

Recently I learned in class Caratheodory's convex hull theorem which suggests that in a finite dimentional (n) space every x can be written as a convex-combination of maximum (n+1) vectors. I was ...
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Is the algebraic interior relatively open in a closed convex set?

I am struggling with proving or disproving the following: Let $X$ be a locally convex space and let $C\subset X$ be closed convex with non-empty algebraic interior which we denote by $C^i$ (recall ...
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40 views

Jensen's inequality: proof by using linear functions

Here's an extract from Stochastic Calculus for Finance Volume 1 by Shreve. I don't understand the statement that says a convex function is the maximum of all linear functions that lie below ...
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Origin of the term `quermassintegral'.

What is the origin of the term `quermassintegral'? I think this is a german word. What would be its literal translation in English? The definition of quermassintegrals from wikipedia: Let ...
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Equivalence of semi concavity of function $g$ and convexity of function $x\mapsto \frac c 2 |x|^2 - g(x)$

$g\in C^2(\mathbb R^n)$ is called semi concave, if there exists $c>0$ such that for all $x,y\in\mathbb R^n$ the following holds: $$g(x+y) - 2g(x) + g(x-y) \leq c|y|^2$$ Now, in Evans "Partial ...
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Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
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How to construct a suitable example in matrix convex

To show that the function $X \to X^{3}$ is generally not matrix convex of order 2 on $S_{+}^{2}$. I cannot find an example and even don't know how to construct one.
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Why a set of positive definite matrices define a half space?

A half-space is a set of the form $\{x|a^Tx \leq b\}$. Also it is stated that the set $\{X\in S^n | z^TXz \geq 0 \}$, with $S^n$ denote the set of symmetric $n\times n$, is a half space$^1$, Can we ...
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21 views

Second order cone with quadratic interpretation

Could you please help me to understand how the second part of the equation (quadratic form) derived form the first one? The basic definition of the second-order cone is: $C = \big\{(x,t) \in ...
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discussing the existence of a convex function

If $g$ is a positive function on $[0,1]$ such that $g(x)$ tends to $\infty$ as $x$ tends to $0$, then there is a convex function $h$ on $[0,1]$ such that $ h \leq g$ and $h(x)$ tends to $\infty$ as ...
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If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then how do we ...
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What is the sub-differential of the separable sum $R(w) = \sum^{d}_{j=1} |w_j|$?

Recall the definition of a sub-differential: $$\partial F(w_0) = \{ v : \forall w, F(w)-F(w_0) \geq v \cdot (w - w_0)\} $$ Intuitively, for any w in the domain of the function one can draw a plane ...
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How does $\in$ behave with simple algebra dealing with sub gradients?

I was trying to understand the following optimization problem: $$argmin_{v \in H} {R(v) + \frac{1}{2}||v - w||^2}$$ Assume $R(v)$ is Convex, proper and semi-continuous with a unique minimizer. ...
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Intuitive explanation of why a chord of a convex function has to be a straight line

I was trying to understand the definition of convexity better. A simple definition of convexity is: $$f(tx_1 + (1 -t)x_2) \leq tf(x_1) + (1-t)f(x_2)$$ $\forall x_1,x_2 \in Domain(f)$ Intuitively, ...
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Strong convexity on sets?

Consider the definition of convex functions: $$ f(tx+(1-t)y) \le t f(x)+(1-t)f(y) $$ It is easy to show the definition of the convexity on sets with respect to the above definition (Specifically for ...
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Strictly convex if and only if derivative strictly increasing?

Suppose $f$ is a real-valued function that is differentiable on an open interval $I$. It is well-known that $f^{\prime}$ is increasing on $I$ if and only if $f$ is convex on $I$. Is the following ...
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Affine functions as equality constraints in convex optimization problems

I am studying on an introduction to convex optimization problems. When defining a convex optimization problem, we have a convex object function, $f(x)$, a set of convex functions $g_i(x)$ where the ...
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32 views

A property of concave functions

If $\phi$ is a concave functions (that is $-\phi$ is convex) with $\phi(1)=0$ why is it that $\phi(x)\le x-1$?
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Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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Convexity and concavity of discontinuous functions

QUESTION F(x) =-x for x>=0 and F(x)=x for x<=0 Is the function convex/(strictly), concave/(strictly) I have attempted the answer but got strictly concave but isnt a discontinuous function meant ...
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27 views

subdifferential and Legendre transform

I have a problem with the following exercise from Evans, Partial Diff. Eq., Chapter 3, problem 6: Let $H:\mathbb R^n\to\mathbb R$ be convex. We say $q$ belongs to the subdifferential of $H$ at $p$, ...
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Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
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Existence of a unique maximizer of a strict quasi-concave function defined over a convex set

Set $S \subset \mathbb R^2$ is compact and convex. A typical element of $S$ is $s=(s_1,s_2) \in S$. Also, $d \in \mathbb R^2$ is a fixed element such that there exists $s \in S$ such that $s \gt d$. ...
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Elementary proof of an inequality with $e^x$ when $|x|<1$.

Assume $|x| <1 $ and we already know $0 \leq e^x - 1 - x$. Note that last inequality was god given, we know it is true, but we do not know how it was proved. Can we deduce from here that $e^x - 1 - ...
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If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$. Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$. I need to show that for $\epsilon ...
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Epigraph of closed convex hull of a function

$\newcommand{\co}{\overline{\operatorname{co}}}\newcommand{\epi}{\operatorname{epi}}$ Let $X$ be an n.v.s and $f: X \to \mathbb{R} \cup \{+\infty\}$ and define $$\co f(x) \doteq \sup_{\substack {x^* ...
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Showing convexity, having trouble showing positive definiteness

I am interested in showing the convexity of $$-\log(-f(\pmb{x}))$$ for $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{-}$ and $f$ convex. If we let $\nabla f$ denote the column vector where the $i$th ...
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If $A$ is a convex set in $R^n$ with a limit point $x_0$, can we have an open line segment in $A$ with $x_0$ its limit point…? [closed]

If $A$ is a convex set in $R^n$ with a limit point $x_0$ outside $A$ can we have an open line segment in $A$ with $x_0$ its limit point...?
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Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
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Confusion about Concavity

Simple question. The function g of a single variable is defined by g(x) = f(ax + b), where f is a concave function of a single variable that is not necessarily differentiable, and a and b are ...
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(still open) For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
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Prove concavity without testing the second derivative..

Consider a function $F(L)=(L-L^2a)T^{L-1}$, where $0<L<\frac{1}{a}$. The constants $a$ and $T$ may take values over $]0,1[$ and $[0.01,0.1]$, respectively. The first derivative of $F$: ...
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About hyperplanes on the boundary (with no $C^1$ regularity ) of compact convex sets

I am reading a paper and the authors use the following property: "Let $K$ a compact and convex set in $R^n$ with nonempty interior. Let $x_0 \in \partial K$ and suppose that the boundary is not ...
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24 views

Confusion with a proof about the continuity of convex functions

I studying convex analysis and in my book I have the following statement and proof: Lets assume that $f:S\rightarrow \mathbb{R}, \;S\subset \mathbb{R}^n$ is a convex function. Then $f$ is ...
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About supporting hyperplanes of convex sets

Let $K \subset R^n$ a convex set, and $x \in \partial K$ such that that there exists a closed ball $B(x_0,R) \subset K$ of positive radius with $x \in B(x_0,R) $. My intuition tells me that there ...
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How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
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Strictly convex set

When a function is strictly convex it has many desirable properties, most notably that it admits a unique minimum. I was wondering if there is anything desirable about a strictly convex set (meaning ...
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Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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Projected Area vs. Surface Area of a 3D Set

(In what follows, I'm making up the nomenclature as I go along, so please pardon anything nonstandard.) Suppose I have a set of points $A \in \mathcal{R}^3$ which is compact, convex, and simply ...
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Optimisation over matrix entries

I was looking to write the KKT conditions to solve this optimisation problem. $$\min_{\substack{\sum_j x_{ij}\le k_i \\ i=1,2,\ldots N}} a^\top (I-X)^{-1} b $$ Since there are $N^2$ decision ...
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Projection onto a matrix where the diagonals are identity matrix

I'm trying the understand intersection of convex sets given in "Convex Optimization - Boyd" which I'm also trying to code in cvx. The two convex sets I'm trying to find the intersection are given ...
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Efficient way to compute the strong convexity modulus of a function?

I have a strongly convex function $f:X\to\mathbb{R}$, where $X\subseteq \mathbb{R}^n$, with strong convexity parameter $\sigma>0$. By definition $f$ satisfies, for all $x,y\in X$ and $t\in[0,1]$, ...
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When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
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A question about convex open set in a topological vector space.

Supose $E$ is a topological vector space(may not be Hausdorff). $U\subset E$ is an open set such that $U+U=2U$. How to show $U$ is convex? I can see if $E$ is $T_1$,then $E$ should be Hausdorff. ...
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Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
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If -log(f) is convex, is f automatically convex?

Say I want to know if $f(x)$ is convex. Can I apply any convex function, strictly increasing function to it and preserve convexity? Say $f(x),g(x)$ are convex and strictly positive and I want to know ...
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an interior point of a convex set

How can we prove a point is an interior point of a convex set, considering we don't have all of the extreme points of the given convex set ? or How can we find an interior point of a convex set, ...
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Hausdorff Distance between Subdifferential sets

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
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Quantitative aspect of Caratheodory theorem

Let A be a compact convex set in n-dimensional space. [ Of principal interest is n > 2 . ] A result of Caratheodory states that A is equal to the union of its simplices (i.e. simplices with all ...
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32 views

checking for convexity/concavity of a function

i'm having some problem in establishing the convexity/concavity of the following two functions. Check for the concavity/convexity of the following functions: (a) ...