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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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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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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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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28 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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relative interior and the affine map

In the convex analysis book by Hiriart-Urruty &Lemarechal, Proposition 2.1.12 states $ri [A(C)] = A(ri C)$. Where $ri$ is the relative interior and $A: \mathbb{R}^n \to \mathbb{R}^m $ is an ...
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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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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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30 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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26 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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51 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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44 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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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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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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57 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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178 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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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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60 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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36 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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60 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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25 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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42 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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43 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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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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43 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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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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55 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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Book/Papers for properties of convex/ uniformly convex Banach Spaces

I am looking for reference books and research articles which cover analysis of uniformly convex and strictly convex Banach spaces.
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19 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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24 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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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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Prove a complex function to be convex

I have a function and want to prove that it is convex when $0 \leq x \leq 1$: \begin{equation} f(x)=\frac{b1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( \gamma+b1) } \end{equation} and \begin{equation} ...
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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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42 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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Cyclical monotonicity

I am deeply troubled by a question for the homework. Either prove or a give a conter-example to the following claim: A continuously differentiable function $f:\mathbb{R}^l\to\mathbb{R}^l$ is ...
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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
59 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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91 views

Computing convex hull of a bunch of circles

I am stuck on the following question ...
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Convex hypograph implies convex level sets. Is this proof complete and correct?

I want to show that when the hypograph of a function is convex, then the upper level sets are convex too. By definition, a pair $(x, a)$ belongs to the hypograph $H$ if $f(x)\geq a$. Let's suppose ...
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Hyperplane optimization for Support Vector Machines

I am trying to learn about the theory behind the Support Vector Machines, by reading the tutorial at: http://research.microsoft.com/pubs/67119/svmtutorial.pdf In its most basic form, SVMs is a binary ...
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$C^2$ 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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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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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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Differentiability of Moreau-Yosida Regularization? [duplicate]

I'm looking for a proof of the differentiability of the Moreau-Yosida regularization of a proper closed convex function $f(y)$ defined on an n-dimensional Banach space $Y$. namely the function is ...
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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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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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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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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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28 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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32 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 ...