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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tangent cone of a projection

I am stuck with what looks like a basic problem from convex analysis and I don't seem to have the answer. Any pointers or insights would be of great help. Suppose $K$ is a closed convex set in ...
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1answer
19 views

Prove that $\log I_{\nu}(x)$ is concave

As the title suggests I need to show that the log of the modified bessel function is concave. When I graph it, certainly seems to be the case. So far I have that: $$ y=\log I_{\nu}(x)\\ ...
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45 views

Describing the minimizers of this function

Consider the following continuous function over $x$, with $a,b>0$: $$f(x)=\begin{cases} ax-\sqrt{x} & \text{for }x\leq b^{2}\\ ab\sqrt{x}-b & \text{for }x>b^{2} \end{cases}$$Note that ...
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43 views

intuitive question about the boundary of a set

Consider $\Omega_1, \Omega_2$ two convex, bounded domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ suppose that $\operatorname{dist}(x, \partial \Omega_2) =\min_{y \in \partial \Omega_2} ...
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1answer
49 views

How to know if a function is not “too convex”?

In my math courses, I have never come across the idea of being "too convex", but this is from an economics course. Essentially, you have some function $P(Q)$, where $Q>0$. The model tells us to ...
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24 views

Why $\frac{\partial u((1-t)x+ty)}{\partial t} \mid_{t =0} \geq 0$, if $u(x) \geq u(y)$ and $u$ is quasiconcave and differentiable?

Let $u$ be quasiconcave and differentiable at $x$. If $u(x) \geq u(y)$, then how to show that $\frac{\partial u((1-t)x+ty)}{\partial t} \mid_{t =0} \geq 0$? $u$ is quasiconcave means that for all ...
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2answers
134 views

Conditional expectation of a random vector taking values in convex sets

on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ i have a random vector $X\in L^1_{\mathbb{P}}(\mathbb{R}^d)$ (integrable with values in $\mathbb{R}^n$), such that $\mathcal{P}-a.s.$ $$X\in ...
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1answer
41 views

Convexity problem

Let $S$ be a convex set. If $x\in$ int$S$ and $y\in$ cl$S$, show that relint[x,y] $\subset$ int$S$. I easily proved this for a case where y is in the interior of S, but am stuck if y is in the ...
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1answer
33 views

Is the function (sum-of-squares) / sum convex on nonnegative input?

Let $$f \colon \mathbb{R}_{> 0}^n \to \mathbb R$$ be defined by $$f(x_1,\dotsc,x_n) = \begin{cases} 0 &\text{if }x_1 = \dotsb = x_n = 0\text{,}\\ \frac{\sum_i x_i^2}{\sum_i x_i} ...
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1answer
187 views

An exercise on convex decreasing function properties

A function f$(x)$ defined for $x\geq0$. It is positive, decreasing, convex and log-convex: $\frac{d^2}{dx^2}\log[f(x)]>0$, $f(0)<1$. Can we prove that $f''(x)x+f'(x)>0$ for sufficiently ...
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52 views

$f:D\subset \Bbb R^2 \rightarrow \Bbb R$, where $D$ is a compact and convex set, reaches it maximum at $int(D)$

I'm trying to prove that if $D$ is a compact and convex (for every two elements of $D$, the line that connects them is contained in $D$) then: If $f:D\subset \Bbb R^2 \rightarrow \Bbb R$ and at ...
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91 views

Convexity of $S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ^{n}\right\}$

i've to prove that the following set is convex: Let $A \in \mathbb{C}^{n \times n}$ $$S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ...
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1answer
155 views

Minkowski sum of two polytopes via the halfspace representation

If i have two polytopes denoted by $P_1, P_2 \subset \mathbb{R}^d$, suppose their halfspace representations are respectively $H_1x \leq K_1$ and $H_2x \leq K_2$. Now, considering their Minkowski sum, ...
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1answer
68 views

How to find the convex hull of a given set?

$A=\{(0,0),(0,1),(1,0)\}$ $B=\mathbb{Q}^2$ $C=\{(x,\sqrt{x})\in \mathbb{R}^2:x\ge0\}$ I have to find Conv(A), Conv(B) and Conv(C). My attempt Conv(A) is the boundary (correction: obviously it ...
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204 views

A short question about the convexity of a function

Let $x$ and $y$ be two numbers; $0\leq x \leq 1$ and $0\leq y \leq 1$ satisfying $$\mathcal{X}\times \mathcal{Y}=\left\{(x,y):\sum^{\lfloor k\rfloor}_{i=0}\binom{n}{i}(1-y)^{i} y^{n-i} +\sum^{\lfloor ...
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39 views

is $R^N_{ ++}$ a convex set?

Is $R^N_{ ++}$ a convex set? I'm working on some optimization hw problems that have some functions of the type: $f:\mathbb{R}^2_{++} \rightarrow \mathbb{R}$ And it seems like in general whenever ...
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166 views

when does the Minkowski inequality for infinity norm become equality

I have two vectors $x, y \in \mathbb{R}^d$, it is well known as Minkowski inequality that: $|x+y|_\infty \leq |x|_\infty + |y|_\infty$, where $|x|_\infty= \underset{i=1..d}{\max} |x_i|$ with $x = ...
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523 views

When is the difference of two convex functions convex?

Assume that $X$ is a finite dimensional Banach space. I know that in general if two functions $f:X \mapsto \mathbb{R}$, $g:X \mapsto \mathbb{R}$ are convex then the function $(f-g):X \mapsto ...
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2answers
486 views

Prove that convex function on $[a,b]$ is absolutely continuous

In the book Roberts, Varberg, Convex functions, on pp.9-10 is proved that If $f:(a,b)\rightarrow \mathbb R$ is continuous and convex then $f$ is absolutely continuous on each $[c,d]\subset ...
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1answer
40 views

Why should the points that define a simplex be affinely independent?

I have a question regarding the definition of a simplex. I am quoting the definition of a simplex from Wikipedia, which is similar to the definition in my textbook too "Specifically, a $k$-simplex is ...
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1answer
116 views

The definition of convex body and the Hilbert cube

I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set ...
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1answer
97 views

Algorithm - the longest chord whose supporting line contains a given point, in a convex polygon

"Let $P$ be a convex $n$-gon and $q$ a point in the plane. Find an algorithm to compute the longest chord whose supporting line contains q." When $q$ is external to $P$, I think I can prove the ...
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2answers
204 views

algorthm to find a farthest point in a convex polygon to an external point

Given a point $q$ external to a convex polygon $P$, propose an algorithm to compute a farthest point in $P$ to $q$. One can always have at least one vertex of $P$ in the set of farthest points of $P$ ...
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1answer
356 views

algorithm to find closest point in a convex polygon from an external point

Given a convex polygon $P$, and a point $q$ of the plane, external to $P$, what is the fastest algorithm to compute the closest point in $P$ to $q$. A linear algorithm of course works, computing the ...
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143 views

Isometric isomorphism maps extreme points to extreme points

I am trying to prove that $\mathcal{c}$ and $\mathcal{c_0}$ are isomorphic but not isometrically isomorphic. I've read on this forum that isometric isomorphism preserves extreme points, but I don't ...
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1answer
233 views

Compact set and its extreme points

I am reading Chapter 3: Convexity of Rudin's "Functional Analysis". Here is the problem I'm having trouble solving (number 18): Let $K$ be the smallest convex set in $\mathbb{ R}^3$ that contains ...
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64 views

Verifying a production set is a convex cone

This comes from a paper that I am reading: For $i=1,2$, suppose that $F_i(\cdot,\cdot)$ satisfies the assumption: $F_i(K_i,L_i)$ is defined for all $K_i\geq 0$, $L_i\geq 0$. $F_i(0,0)=0$. ...
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1answer
38 views

Will continuous extension preserve strict convexity?

The problem I am thinking about is like follows. Suppose that $h$ is a strictly convex function on an open convex set $S$. Then, we extend $h$ continuously to the closure of $S$ that is denoted by ...
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1answer
99 views

Uniqueness of projection in a Banach space

Let $X$ be a Banach space, $M$ be a subspace of $X$ and $x \in X$ be any vector in $X$. Consider $\displaystyle \hat{x}_M=\arg \inf_{m\in M}\|x - m\|$. Under what conditions for $l_p$ norms $p = ...
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133 views

Minimum distance from a subspace in a Banach space

$X,Y,Z$ are disjoint sets of vectors in a Banach space M. $S_X,S_Y,S_Z$ denotes the subspace formed by the vectors in $X, Y$ and $Z$ respectively. $S_{YZ}$ and $S_{XZ}$ denote the subspaces formed by ...
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1answer
75 views

Question about the structure of a convex optimization problem

I am reading a tutorial about Convex Optimization and it defines the general Convex Optimization problem as: $$ minimize_x f(x)$$ where $g_1(x) \leq 0, ... , g_m(x) \leq 0$ and $Ax=b$ and $x \in ...
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73 views

Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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1answer
385 views

Onion-peeling in O(n^2) time

I am working on the Onion-peeling problem, which is: given a number of points, return the amount of onion peels. For example, the one below has 5 onion peels. At a high level pseudo-code, it is ...
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1answer
143 views

Computing convex hull of a bunch of circles

I am stuck on the following question ...
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1answer
102 views

$C^2$ approximation of a convex set with a “flat part”

Suppose we have a closed, bounded, convex set $K \subset \mathbb{R}^n$ with non-empty interior. It's well-known that we can approximate $K$ either from the inside or from the outside in the Hausdorff ...
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1answer
250 views

Sufficient condition for global maximum of strictly quasi-concave functions (unconstrained)?

Suppose $f(x)$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and $f$ is strictly quasi-concave. If $x^*$ is a point such that $f'(x^*)=0$, then can we say that $x^*$ is a global maximum of this ...
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1answer
1k views

A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, ...
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577 views

Convexity of Matrix Exponential

Consider the function $A: \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ defined as $$ A( x ) := \left[ \begin{matrix} x_1 & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & x_2 & ...
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1answer
52 views

Is the closure of a geodesically convex set convex?

My question is Is the closure of a geodesically convex set convex? If so, is there a simple proof for it? In $ R^n $ there is a simple proof for it through convergent sequences. How should I apply ...
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1answer
40 views

Convexity of an exponential function

I failed the following question in a quiz: For which values of $a$ the function $e^{-a\sqrt(x)}$ with $dom = \mathbb{R}^+$ is convex? Check all that apply: $a\leq0$ $a\geq0$ $-1 \leq a\leq1$ ...
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1answer
242 views

what does full- dimensional means when speaking about covex cones

I want to know the exact definition of full-dimensional. And what does "dimension" refer to, is that in the sense of algebraic variety? I have read several writing announcing that the cone of ...
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1answer
130 views

Convex Sets Pre-image

I am struggling with the following question: Let $a \in \mathbb{R}^n $ and $ b \in \mathbb{R}$ and define $ f: \mathbb{R}^n \rightarrow$ $\mathbb{R} $ by $f(x) = \langle x,a \rangle + b, x \in ...
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1answer
58 views

Proofing set to be convex

I am struggling solving the following exercise: Let $a \in \mathbb{R}^n$ and $ b \in \mathbb{R}$ and define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by $f (x)=\langle x,a \rangle + b, x\in ...
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1answer
122 views

Example of a convex set whose closure is not convex?

An enumeration $ν\colon ℕ → A$ of the rationals $A$ in $(0..1)$ yields an open set $U_ν = \bigcup_{k ∈ ℕ} B_{1/4^k}(ν(k))$, containing all of $A$. You can choose $ν$ such that $U_ν ⊂ (0..1)$ (by using ...
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1answer
51 views

Compactness, Convexity, Convex Hull of Sets including sequences

Is the following set compact, is it convex and what is the convex hull? $V = \{(x_1, x_2,...,x_n) \in \mathbb{R}^n :\frac{1}{1 + i} \leq x_i \leq \frac{1}{i}, i=1,2,...,n\}$ My thoughts: I was ...
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1answer
83 views

Consider the problem minimize $f(x_1,x_2) = (x_2 −x_1^2)(x_2 −2x_1^2)$

(i) Show that the first- and second-order necessary conditions for optimality are satisfied at $(0,0)^T$. (ii) Show that the origin is a local minimizer of f along any line passing through the origin ...
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2answers
74 views

Quasiconcavity of incomplete gaussian integral

From visual experiments , it appears that the set $$ S_r = \left\{ (x,y) \text{ s.t. } \int_x^y e^{-t^2} dt \geq r \right\} $$ is convex for $r \geq 0$. Or equivalently, the function $$ f(x,y) = ...
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3answers
185 views

Why doesn't the definition of the interior of a set depend on the dimension of the set

I have just started with a course on convex optimization and have been introduced to the concept of the interior of a set. I have a fairly basic question. I am still trying to understand this topic, ...
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2answers
1k views

Is every convex function on an open interval continuous?

Let $f:(a,b)\rightarrow \mathbb{R}$. $f$ satisfied the following property: If $\forall x_{1},x_{0},x_{2}\in(a,b)$ and $x_{1}<x_{0}<x_{2};$then$\frac{f(x_{0})-f(x_{1})}{x_{0}-x_{1}}\geq ...
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1answer
83 views

Dual of concave function is convex

If $U(x)$ is strictly increasing and strictly concave and $lim_{x \rightarrow \infty}$ U'(x) = 0, prove that its dual: $$U^{*}(y) = max_x \{U(x) - xy\}$$ is convex. Does anyone know how to prove ...