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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Prove the concavity of the transformation from a concave function to another

Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, ...
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16 views

Convex signal reconstruction for convex generator function?

Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to ...
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73 views

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
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Is this Function of Product of variable and Ratio of CDF and PDF of Standard Normal Distribution Convex?

Let $G\left(x\right)=x\frac{\phi\left(x\right)}{\Phi\left(x\right)}$. Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. Is $G\left(x\right)$ convex? It has been shown ...
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52 views

Jensen inequality for convex functions - infinite countable number of weights

Does Jensen inequality for convex functions hold if there is countable infinite number of weights? For example weights are given by sequence $(\frac{1}{2^n})_{n\in\mathbb{N}_1}$ ? If not, where is ...
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20 views

Convex functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
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26 views

How do I show that the set $S$ in $\mathbb R^3$ defined by linear inequalities is a $3$-simplex?

Consider the set $S$ in $\mathbb R^3$ defined by the inequalities: $x+y+z \ge 1$ $-x+y+z \le 1$ $x-y+z \le 1$ $x+y-z \le 1$ How can I show that $S$ is a $3$-simplex ? (Convex hull of $3 + 1$ ...
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18 views

An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...
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48 views

Minimizing the max function

Suppose we have the single-variable function $$f(x) = \max_k \{f_k(x)\}$$ where each $f_k$ is convex and smooth (and known beforehand). We want to minimize it over some bounded interval. We can, in ...
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Find $U \subset D $ with $|U|$ minimal s.t $v = (1/2,1/2,1/2)$ belongs to the convex hull of $U$.

Consider the convex hull $\text {conv} \{e_1,e_2,e_3,(1,1/2,1/2),(1/2,1,1/2), (1/2,1/2,1)\}$ in $\mathbb R^3$ and the vector $v = (1/2,1/2,1/2)$. I want to compute $U \subset ...
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38 views

Function dominated by convex function is eventually convex

Suppose that we have a twice-differentiable function $f$ on $x\in [0,\infty)$ such that $f(x)>0$ on $x\in [0,\infty)$ (i.e. strictly positive) $f'(x)<0$ on $x\in [0,\infty)$ (i.e. strictly ...
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24 views

Every convex function is locally Lipschitz ($\mathbb{R^n}$)

I know that if $f$ is convex function so $f$ is continuous. And I know too that partial derivatives exists. What can I do?
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98 views

How do I show that $B$ has an infinite number of extreme points, but no faces (sides) of dimension $1$ or $2$?

Let $B = \{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 + z^2 \le 1\}$. How do I show that $B$ has an infinite number of extreme points, but no faces (sides) of dimension $1$ or $2$ ? I've been thinking ...
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2answers
118 views

Prove a closed ball in $\mathbb R^3$ has an infinite number of extreme points.

How do I show that a closed ball in $\mathbb R^3$ has an infinite number of extreme points ? (Closed ball is written as $S = \{(x,y,z) \in \mathbb R^3 | \sqrt {x^2 + y^2 + z^2} \le R \}$) I know ...
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10 views

Convex combination of quasiconvex functions.

Is a convex combination of two quasiconvex functions necessarily quasiconvex? If not, what can be said about the convex combination?
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26 views

Let a polyhedron $P = \text{conv}(S)$ where $S$ extreme points. Can $S' \subset S$ (proper) be a generator?

Let $P$ be a polyhedron and let $S=\{ v_1, \ldots, v_r\}$ its extreme points. Suppose further that $\text{rec}(P)={0}$ so $P=\text {conv}(S)$. How do I see that I cannot remove any points from $S$ and ...
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1answer
48 views

Let $S = \{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove $(1-\lambda)x + \lambda y \in S$ for $x=\lambda'y$, $\lambda' < 0$.

Let $S = \{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. I've verified that $x,y \in S$ implies $(1-\lambda)x + \lambda y \in S$ when $x,y$ are linearly independent using Pythagoras and when ...
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Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these.

Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these. I want to compute all the extreme points of the set $P$ (polyhedron) in $\mathbb R^3$ ...
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Can $\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$ be reduced to a convex hull of a subset of these?

How do I see whether $$\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$$ can be reduced to a convex hull of a subset of these vectors? That is, if $D = \{ e_1,e_2,e_3, ...
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Iterative algorithms to approximate unknown multivariate convex real-valued functions by a set of linear upper/lower bounds

I am looking for iterative algorithms that can approximate a multivariate convex real-valued function $f(\vec{x})=y,\, \Bbb{R}^n\rightarrow \Bbb{R}$. The function is not known beforehand, but it is ...
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5 views

Suppose $P = \text{conv}(\text{ext}(P))$. Is it possible to find $S \subset P$ of cardinality $< \text{ext}(P)$ s.t $\text{conv}(S) =P$?

Suppose that $P$ is a polyhedron and that $P = \text{conv}(\text{ext}(P))$. Suppose further that $ \text{ext}(P)$ has cardinality $r$. How do I see that I cannot find a set $S \subset P$ of ...
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38 views

Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
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Estimate the volume of a convex body given a uniform random sample of points inside it?

Let $K$ be a convex, full-dimensional, bounded region of $R^n$. More precisely, there exist two balls of radiuses $0<r<R$ such that the ball of radius $r$ is fully contained inside $K$ and the ...
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Is the ratio of a decreasing function and an increasing function, a quasi-concave function?

$f(x)$ is a strictly decreasing function and $g(x)$ is a strictly increasing function and positive. Is $h(x) = f(x)/g(x)$ quasi-concave?
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Convexity of the ratio of the standard normal PDF by its CDF

Is there some way to show that the following function $\psi$ is concave or convex? Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. ...
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Let $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove existence of a unique point $v_0 \in B$ that is closest to $v \notin B$?

Suppose $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Consider a point $v \notin B$. How do I prove that there exist a unique point $v_0 \in B$ that is closest to $v$. Also, how do I ...
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Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
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Prove the existence of minimal height of a convex polygon

Suppose we have a polygon in $\mathbb{R}^n$. Obviously, we can always trap the polygon into two parallel hyperplanes perpendicular to any given direction. Now the question is, how to prove the ...
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1answer
89 views

Convex Function

$f: U\subset\mathbb{R}^m \to \mathbb{R}$ is a convex function if $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$, for all $x,y \in U$ and all $t \in [0,1]$. If $f$ is convex and continuous function, and $f$ has ...
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The convex hull of every open set is open

Let $X$ be a topological vector space. Prove that the convex hull of every open subset of $X$ is open. I tried using definition of Convex Hull and Open Set, but I couldn't prove the statement.
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$U$ bisected by all hyperplanes $\implies$ $U$ symmetric?

Let $U \subset \mathbb{R}^n$ be a bounded open convex set, such that every hyperplane passing through the origin divides $U$ into two sets of equal volume ($n$-dimensional Lebesgue measure?). ...
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Showing that $f$ is convex given that $f(\frac{x+y}2)\le\frac{f(x)+f(y)}2$

Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that $f$ is convex. Thanks. Edit: I'm sorry for all the confusion. $f$ is assumed to be continuous on an ...
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Proof that f is convex

Consider $D\subset\mathbb{R}^n$ a convex set and $f_i:D\rightarrow\mathbb{R}$ convex functions in $D$, $i \in I$ is a any set of indexes. Suppose there is $\beta \in \mathbb{R}$ such that ...
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23 views

Multivariate convex function / increasing differences

$\newcommand\Rr{\mathbb{R}}$I am trying to show the following statement. It feels true to me, but I haven't found any reference in the literature so far: Let $\Rr^n$ be ordered component-wise, i.e., ...
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39 views

Is a maximum function concave?

A function such as $$\max (X, Y)$$ given $X\geq 0$, $Y \geq 0$. Because It would form an inverse L shape on the graph, and that looks like a concave function. However my teacher said that the function ...
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Maximal eigenvalue is convex function

Let $A$ be a symmetric real matrix. let $f(A)=\lambda_{max}(A)$ be it's largest eigenvalue. Why is $f(A)$ convex?
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Boundary of convex set is piecewise $C^1$

Let $K$ be a convex and compact subset of $\mathbb{R}^2$. Is it true that the boundary of $K$ can be parameterized by a piecewise $C^{1}$ application $\gamma :I\to\mathbb{R}^2$?
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Convexity of $f(x) = \sum_i x_i^p$ for $x_i \ge 0, p \ge 1$

Let $x = (x_1,\ldots, x_n)$ be nonnegative real numbers and $p \ge 1$, then the function $f(x) = \sum_{i=1}^n x_i^p$ is convex, the following proof is wrong, or not? I have it from here, but then ...
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Orthogonal projection of an $n-$vector onto the subspace ($m\leq n$)of $\mathbb{R}^n$ containing a convex polytope

Lets say we have an $n \times m$ matrix $A$, whose column vectors are $(\vec{\mathbf{0}},a_1,a_2,...a_j)$ are points in $\mathbb{R}^n$ and the non-zero vectors have unit length. Let ...
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Differentiability of support function (even for non-convex)

I am reading an economics book (for those who are interested, MWG Microeconomic Theory) and there's a theorem that was just given without proof, but I am interested in the proof - also because I ...
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199 views

Proof of $Ax = b, x \ge 0$ is a closed subset

I'm trying to follow the The Farkas-Minkowski Theorem (Internet Archive) but I'm having a little bit of difficulty. On the second page the author states, Then we consider a set of the form $R_k ...
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how can we check convex or nonconvex feasible?

example if i have 20 constraints functions.These functions cut the objective function and create the feasible region. Their intersections can become edges and create a nonconvex feasible region even ...
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28 views

Optimising a piecewise convex function

I have a function $f(x)$ defined for $x\in \mathbb{R}^+$ that is decreasing, piecewise convex and continuous. The 'pieces' of $f$ are exponential and the rate of decrease for each piece reduces as we ...
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Quasi concavity and Quasi Convexity-intuitive understanding

I'm having trouble grasping the concept of quasi concavity and quasi convexity. My textbook states that if f is quasi-concave, then f (λx + (1 − λ) y) ≥ min {f(x), f(y)} . Also that is f is quasi ...
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Given Convex Function, Conditions when Variable times Convex Function is convex

Given that say, $f(x)$ is convex for $x>0$. We can arrive at the following conditions for when $xf(x)$ would be convex. Please add anything that I might have overlooked and further simplifications ...
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Is an lsc sublinear function $X^* \rightarrow (-\infty, \infty]$ always a support function for some closed non-empty $C \subset X$?

I can't seem to find any resources on this, even though it seems like an obvious question to ask. The separation theorem implies that, if we have an lsc sublinear function $\phi : X^* \rightarrow ...
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1answer
46 views

Isn't every increasing continuous convex function strictly increasing (disregarding $f(x) \equiv 0$)?

Isn't every increasing continuous convex function $f$ strictly increasing (disregarding the trivial case $f(x) \equiv 0$)? I don't see any counterexample!
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60 views

Proof that Convex Function with alternate variable is convex

Given that say, $f(x)$ is convex for $x>0$. Would the proof that $f(a-x)$ where $a>x$ is convex proceed as shown below. Please verify the correctness and/or alternative approaches that can ...
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18 views

Properties of a proper cone

Let $K$ be a proper cone. I need to prove following properties: if $x \preceq_K y$ and $u \preceq_K v$, then $x+u \preceq_K y+v$ if $x \preceq_K y$ and $y \preceq_K z$, then $x \preceq_K z$ if $x ...
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25 views

Is a twice differentiable function whose only extrema is a minimum automatically convex?

I have a twice differentiable function $H(x)$ for which I have already proven that: $\exists !x^*:H'(x^*)=0$ and furthermore that $H''(x^*)>0$ (the only extremal is a minima). $H''(x)$ is ...