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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Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
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Showing a quotient $\mathbb{Z}$ module is free

In Fulton's "Introduction to Toric Varieties" he repeatedly uses the following fact. Let $\sigma$ be a strongly convex rational polyhedral cone in a lattice $N$ and let $N_{\sigma}$ be the subgroup ...
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Volume of the projection of the unit cube on a hyperplane

Let $C_n\subset\mathbb{R}^n$ be the $n$-dimensional cube with side $1$, and let $P_k$ be any $k$-dimensional plane, $k\leq n$. What is the maximal $k$-volume $V_{n,k}$ of the projection of $C_n$ on ...
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Proof of an obvious fact about convex polygons.

Consider a closed simple polygon in the plane. It is intuitively obvious that the polygon is convex if and only if all the interior angles measure less than or equal to $\pi$ radians. I have never ...
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Is there a geometric argument that the Legendre transform of a convex function is convex?

I am trying to build intuition on Legendre transforms. Arnold's Mathematical Methods of Classical Mechanics has some nice geometric interpretations, but he does not provide a proof that the Legendre ...
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Hausdorff distance via support function

Let $A$ and $B$ be convex compact sets in $\mathbb{R}^n$. Define $$ h_{+}(A,B) = \inf \left\{ \varepsilon > 0 \mid A \subseteq B+\mathbb{B}_{\varepsilon} \right\} $$ where ...
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Is this function convex when the input vector is positive?

I am wondering if $f(\mathbf{x})$ is convex on the input of a vector of $n$ positive reals $\mathbf{x}$: $$f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$$ where ...
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Convexity of inverted sum of positive definite matrices

I am currently working with a class of functions, where every function looks like $f(x)=v^T(xA+(1-x)B)^{-1}v,$ where $v\in\mathbb{R}^n$ is an arbitrary vector, $A,B \in \mathbb{R}^{n\times n}$ are ...
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Differentiability of Moreau-Yosida approximation.

I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is ...
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863 views

Proving Helly's theorem

The problem is to prove Helly's theorem for the case, when the convex bodies are polytopes, by using linear programming duality. Helly's theorem Let $B_{1},...,B_{m}$ be a collection of convex ...
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Describing a set of normals

In a finite dimensional (think Euclidean) ambient space, let $S$ a compact, convex set and $x$ not in $S$. The two sets can be (weakly) separated, i.e. there exists a vector (normal) that defines a ...
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Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
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does this convex set have a specific name?

Let $x_1,\dots,x_N$ be points of $\mathbb{R}^n$. Define the following set: $\mathcal{A} = \left\{\sum_{j=1}^N a_j x_j : -1 \le a_j \le 1, \, \, \forall j=1,...,N\right\}$. It is an easy exercise to ...
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When the closure of a convex set contains a ball

Suppose $C$ is a convex set in $\mathbb{R}^n$ whose closure contains the open ball $B(x,r)$. Is it true that $C$ contains $B(x,r)$? Motivation: I am asking this because something like this seems to ...
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Product / GM of numbers, with fixed mean, increase as numbers get closer to mean.

I am trying to prove a statement which goes like this. Let $a_i$ and $b_i$ be positive real numbers where $i = 1,2,3,\ldots,n$; where $n$ is a positive integer greater than or equal to $2$, such ...
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Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
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Convexity of Exponential Composite Function

$f:\mathbb{R}_{+}^M\rightarrow\mathbb{R}_+$ is a convex analytic function. For $\mathbf{x}\in\mathbb{R}_{+}^M$ and $y\in\mathbb{R}_{+}$, consider the function ...
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strict convexity with a measure theoretic property

Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
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$\int_0^1 u(t)\phi''(t)dt \geq 0,\ \forall \phi\in C_0^1((0,1)), \ \phi\geq 0$. Is $u$ convex?

Suppose that $u\in C([0,1])\cap C^1((0,1))$ satisfies for all $\phi\in C_0^2((0,1))$, $\phi\geq 0$ $$\int_0^1 u(t)\phi''(t)dt \geq 0$$ Can we conclude that $u$ is convex? Note: $C_0^2((0,1))$ is ...
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Convex Alternatives to the Gamma Function?

The Bohr-Mollerup Theorem states that the gamma function is the unique function $f: (0, \infty )\rightarrow \mathbb{R}$ satisfying $f(1)=1,$ $f(x+1) = x f(x),$ and the condition that $\log f$ is ...
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Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
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Visualizing Euclidean Algorithm in $\mathbb{Q}(\sqrt{-7})$ and $\mathbb{Q}(\sqrt{-11})$ with Convex Geometry

In an attempt to answer one of the bounty questions, I have started picturing Euclidean division in quadratic fields. In theory we would like the equation: $$ a = b\,q + r ...
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Is a convex salient cone necessarily contained in an open half-space?

A cone $C$ in $\Bbb R^n$ is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if $C \cap (-C) \subset \{0\}$. Obviously, a cone $C$ such that that ...
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Prove a certain property of linear functionals, using the Hahn-Banach-Separation theorems

I've been trying to solve this question, with no luck so far: Let $X$ be a real linear space, and $\{\|\cdot \|_i\}_{i=1}^{n}$ family of norms on $X$. Let $f$ be a linear functional on $X$ such that ...
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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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Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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Number of supporting hyperplanes

I know that, for any convex set $S$, there is at least one supporting hyperplane at every point in $B$, the boundary of $S$. Also, there can be more than one supporting hyperplane at the same point in ...
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Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
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On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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Convexity of polylogarithms

I want to prove the following proposition: The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$. And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
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Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
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Is there a notion of 'pre-convex' sets?

A set $A$ in $\mathbb{R}^n$ is convex if for any two points $p,q \in A$ and real $\lambda \in [0,1]$, the point $\lambda p + (1-\lambda)q$ is also in $A$. There are many beautiful theorems about ...
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Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
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Cyclically monotone sets on four points

A subset of $\mathcal{X} \times \mathcal{Y} \subset \mathbb{R}^d \times \mathbb{R}^d$ is cyclically monotone if $\sum_{i=1}^n \langle x_i,y_i\rangle \ge \sum_{i=1}^n \langle x_i,y_{i+1}\rangle$, where ...
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Convexity of a Given Function

Is the following function convex or concave? $$f(\mathbf{x}) = \frac{1}{N}\sum_{i=1}^N\sqrt{1+\frac{x_i^2}{\left(\frac{1}{N}\sum_{i=1}^{N}x_i\right)^2}},$$ $\mathbf{x} = [x_1,x_2,...,x_N]^T, x_i \ge ...
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Convexity of cubic vector function?

First question in these forums so go easy on me. I have a function $f_i(x):\mathbb{R}^N\to\mathbb{R}$ which is defined by $$f_i(x) = \frac{(x^TAe_i)x^TAx}{(x^TA+b^T){\bf 1}}$$ where ...
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Find optimal measure

Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem $$ \int\limits_{\Omega}f(x)\,\mu(dx) \to ...
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Prove one set is a convex hull of another set

Define two sets: $A = \{x \in \{0,1\}^n : \lVert x \rVert_1 \leq k\}$ is a finite set of binary vectors; $B = \{x \in [0,1]^n : \lVert x \rVert_1 \leq k\}$ is an infinite set of real-valued vectors, ...
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A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
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Are convex hulls of closed sets also closed?

If $X$ is a compact set in $\mathbb R^n$, how can we show that $\operatorname{conv} X$ is compact as well? Can we say something similar without assuming boundedness, i.e., are convex hulls of closed ...
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Show by example that the Minkowski sum of two sets $X+Y$ may be convex even if neither $X$ nor $Y$ are convex

There were two parts to this question. I proved that the Minkowski sum of two sets $X+Y$ is convex whenever $X$ and $Y$ are convex, but how do I prove this second part? "Show by example that the ...
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Is second derivative of a convex function convex?

If $f$ is twice differentiable and convex, is it true that $f''$ is a convex function ?
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How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?
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Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
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How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $y \gt 0$?

How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $ y \gt 0$? I take the Hessian matrix of $\displaystyle \frac{x^2}{y}$, and I got: $$H = \displaystyle\pmatrix{\frac{2}{y} & ...
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Proving $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$

Let a,b,x,y be positive reals. Prove $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$ I don't have any olympic background, so I may be missing some standard trick. The ...
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Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
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Using the definition of a concave function prove that $f(x)=4-x^2$ is concave (do not use derivative).

Let $D=[-2,2]$ and $f:D\rightarrow \mathbb{R}$ be $f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative). Attempt: ...
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Convexity of intersection

I have been asked to prove that, given a convex set $C$, its intersection with a line is also convex. From convexity definition, I have that $\forall x_1,x_2\in C, \alpha x_1+\beta x_2 \in C$ with ...
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Euclidean Metric and Convexity

Question: Consider the Euclidean metric space $(\mathbb{R}^n , \Vert\cdot \Vert)$. Let $X\subset \mathbb{R}^n$ and $f\colon X \to \mathbb{R}$. $X$ is said to be a convex set if for every $x,y \in X$ ...