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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An inequality regarding convex functions

For a convex function $f(\cdot)$ for $x_4 \ge x_3 \ge x_2 \ge x_1 \ge 0$ and $t,t^{'}\ge 0$ we are given that \begin{equation} \frac{f(x_4)-f(x_3)}{f(x_2)-f(x_1)} \ge ...
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Proof regarding convex sets

A set of points is said to be convex provided that every pair of points in the set can be joined by a line segment that lies entirely within the set. Show that, if $ | ∇f(x)| ≤ M \space \space ...
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Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
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Two fundamental questions about convexity of a function (number2)

The second question is as follows (the first one is here): Consider a function of two variables $f(x,y)$ on some compact domain. Let it be convex in $x$ and concave in $y$. Is it true that ...
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Two fundamental questions about convexity of a function (number1)

The first question is as follows (see the second one): If $f$ is a convex function it is know that there is at most a single minimum. However the argument of the minimum is not guaranteed to be ...
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Convex downward function and its inverse function

How to prove that if function $f$ is convex downward and invertible then $f^{-1}$ is convex downward or convex upward? When is it downward and when upward?
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Polynomial and convex functions

Consider polynomials $\mathbb{R} \rightarrow \mathbb{R}$. I have to Give an example of polynomial that isn't convex downward nor convex upward. Give an example of polynomial that is convex downward ...
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Lipschitz constant of the convex function $f(x) - \frac{a}{2} |x|^2$

I was going through this blog post https://blogs.princeton.edu/imabandit/2013/04/04/orf523-strong-convexity/ It has been mentioned without proof that for a function ...
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Function going from $0$ to $1$ with minimal concavity

How small can I take $C>0$ such that there exists an $f\in C^2(\mathbb R;\mathbb R)$ satisfying the following properties: $f(x)=0$ for $x\leq 0$ and $f(x)=1$ for $x\geq 1$ $f'(x)\geq 0$ for ...
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Does every boundary segment of a convex polyhedron lies on one of its faces?

When I was reading this note, I found Theorem 3.1.5 said: Let $P\in\mathbb{R}^n$ be a polytope whose affine dimension is $d$. Then, every point on the boundary of $P$ lies in a facet of $P$. I have ...
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condition for uniform input to minimize convex function

Let $f:\mathbb{P}(X) \rightarrow \mathbb{R}$ be a convex function where $\mathbb{P}(X)$ is the set of probability distribution on (finite) set $X$. I am looking for condition on $f$ so I can said ...
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Convex hull of convex set boundary

A is a closed convex set with non-empty interior. Does A must equal to the convex hull of its boundary? I know this is false when A is half space. But what about other sets?
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convergence of infimal convolution

I am trying to show the following statement: Let $f:\mathbb{R}^n\to \mathbb{R}\cup\{\infty\}$ be a convex function. Let $f_\epsilon(x)=\frac{|x|^2}{2\epsilon}$. Show that $\lim_{\epsilon\to ...
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How can I solve $\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$ in a closed form depending on projection?

I'm trying to solve the minimization problem $$\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$$ where $b \in \mathbb{R}^n$, $X \subset \mathbb{R}^n$ is a convex set. And $A$ is a symmetric ...
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Can I assume $g$ is finite for proof involving infimal convolution

I am trying to show the following statement: Let $f,g:\mathbb{R}^n\to \mathbb{R}\cup \{\infty\}$ be two convex functions. Assume that there are constants $C_1,C_2>0,\alpha>1$ such that ...
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If f(x, y) is convex, is g(x)=f(x, c) convex, for any constant c?

If $f(x, y)$ is convex (concave) defined on $\mathbb{R}^2$ and $g(x)=f(x, c)$, $c\in \mathbb{R}$, then is $g(x)$ necessarily convex (concave)?
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Are these equivalent definitions of faces of convex sets?

I consulted several books and found that the definitions of faces and exposed faces of convex sets are a bit messy. Many books just treat them as the same stuff. In book "foundations of ...
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Are these inconsistent definitions of extreme subset of convex set?

I need some elementary knowledge of polyhedral and I am consulting several books. However, I found the definitions may not be consistent. I am not sure if I understand them right. The question is ...
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Does every LCS has a convex balanced local base?

Does every LCS--locally convex (topological vector) space has a convex balanced local base? Then it implies every LCS is topologized with a countable number of separating seminorms. So there seems ...
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Proof of Jensen's inequality - why is this progression valid

I'm reading a proof of Jensen's inequality in the following version: Let $f: I\rightarrow\Bbb{R}$ be a convex function defined on an interval $I$. Then: $$\forall{x_1,...,x_n\in ...
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Convex Optimization: Advantages of Symmetric Primal-Dual Algorithms?

This is a follow-up to an answer on a previous question on PD algorithms: http://math.stackexchange.com/a/1193928/36257 I have done some research learning the mechanics of how Infeasible PD ...
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“No cone of this fan contains a nonzero linear space”

On page 393 of this paper, Speyer and Sturmfels produce a quotient of a particular geometric object (which happens to be a polyhedral fan), and claim that, as in the question title, "no cone in this ...
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Geometric description of composition of two convex polygons? [duplicate]

The composition was defined as follow: $(a,b)$ $\in$ $(R;S)$ $\Leftrightarrow$ $\exists c | (a,c) \in R \\ $&$ (c,b) \in S$ . Also the composition was defined as the binary product of two ...
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Is $\mathbf{y}^*$ a local minimizer of $f(\mathbf{h}(\mathbf{y}))$?

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $\mathbf{x}^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible ...
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Determine if $\mathbf{y}^*$ is a local minimizer of $f(\mathbf{h}(\mathbf{y}))$ [closed]

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $x^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible function ...
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How can I solve $y \in (N_X + \nabla f)(x)$ via projection?

I a aware that if I'm trying to solve for $x$ the problem $y \in [\lambda I + N_X](x)$ where $y$ is a known vector, and $N_X$ is the normal cone given by $N_X(x) = \{u : \langle u, x - y\rangle ...
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How to characterize the boundary of a convex set?

I am working on a part of a paper related to topological properties of boundary points. It is important to me to realize the topological and algebraic behavior the boundary points of convex sets. I ...
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Covering of convex sets

This is basic geometry question: Let $a,b,c,d,e$ be five distinct points in $\mathbf{R}^3$ and denote the convex hull of $X$ with $\mathrm{conv}(X)$. How can I prove formally that ...
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Is this sequence of concave functions unbounded?

Let $h_1, h_2,$ etc. be a sequence of positive real numbers such that $$\sum_nh_n = \infty.$$ Let $x_1, x_2,$ etc. be a sequence of real numbers in $(0, 1)$. Let $f_0, f_1,$ etc. be of sequence of ...
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Additional assumption to have a convex image

Let $f\colon \mathbf{R}^2 \to \mathbf{R}^2$ a continuous injective function. In general, it is not true that the image $f[X]$ is convex whenever $X \subseteq \mathbf{R}^2$ is convex. Is there some ...
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Sign of difference of two convex functions

Suppose you have two continuous, convex functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$. Suppose that both $f$ and $g$ are minimised at $x=0$ with $f(0)<g(0)$. ...
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Maximization of probability that all packets are successful simultaneously

I have packet streams $1...k$ and, streams with Prob(err) $p1...pk$. The $p$'s are consts $>0$. I'd like to maximize the probability all make it simultaneously while I'm allowing at most $N$ ...
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Non-empty interior of the convex hull of $f(a),f(b),f(c)$

Let $a,b,c$ be three points in the plane which are not collinear. Let $f\colon \mathbf{R}^2 \to \mathbf{R}^2$ be a continuous injective function. Show that the interior of $$ ...
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Why is this an nuclear norm?

The following figure is from a paper: "The Convex Geometry of Linear Inverse Problems" Just on p.3. As in the figure, the red lines are $2 \times 2$ symmetric unit-Euclidean-norm rank-one ...
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Examples of uniformly convex function

$f$ is uniformly convex with modulus $\phi$ if for any $\alpha \in [0,1]$ and any $x,y$ in the domain, $$f(\alpha x + (1-\alpha)y) \le \alpha f(x) + (1-\alpha) f(y) - \alpha(1-\alpha)\phi(\|x-y\|)$$ ...
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Show that a specific cone in the plane has two generators

Let $u \in \mathbb{R}^2$ and $\{v_1, \dots, v_s\} \subseteq \mathbb{R}^2$ ($s \geq 2$) such that $\{v_i, v_j\}$ is linearly independent if $i \neq j$ and $(u, v_i) > 0$ for all $i \in \{1, 2, ...
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Blackwell's informativeness criterion

Let $a=(a_1,\dots,a_i,\dots,a_n)$ be a probability vector, i.e. $\forall i: a_i\ge0$ and $\sum_i a_i=1$. Suppose $b$ is another $n$-dimensional probability vector. Is it true that there always ...
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Convex Optimization: do Primal Dual methods need to start with strictly feasible point?

I'm learning about Primal-Dual interior point algorithms from Boyd & Vandenberghe Convex Optimization, ch.11.7. Now the text mentions: In a primal-dual interior-point method, the primal and ...
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Convexity of a subset of $R^3$ based on its bounding surface

Let D be a closed region in $R^3$ whose boundary S is a closed, smooth, orientable surface. The tangent plane at every point on S intersects it in a connected set $M$. Is D (the interior of S) a ...
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For convex, smooth function with $\phi''(1)>0$ we have $\phi(s)-\phi(1)\geq \phi'(1)(s-1)+c(s-1)^{2}$

For convex, smooth function $\phi:\mathbb{R}_{+}\to \mathbb{R}$ with $\phi''(1)>0$ we have $\phi(s)-\phi(1)\geq \phi'(1)(s-1)+c(s-1)^{2}$ for constant $c>0$. Attempt By Taylor's theorem ...
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Convex function on closed interval: boundary points?

$I=[a,b]$, let the function $f:I\rightarrow\mathbb{R}$ be convex. (1) Is it possible to prove the existence of the limits: $$\lim_{x\rightarrow a^+}f(x) \ \ \ \ \ \lim_{x\rightarrow b^-}f(x)$$ If ...
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Dense face in Schwartz space

I have a lema and an example for which i don't understand how are they not in contradiction. I am not sure is there something obvius that i am missing. Any help would be much appreciated. Here it is: ...
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Log-Determinant Concavity Proof

Can you please help me understand how he gets the equation marked by red from the above one ?
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Non-empty intersection with the interior of a convex set

Let $C$ be a convex set in the plane with non-empty interior, and let $x \in C$ be a point on its bourdary. Prove that, for each open neighborhood $N$ of $x$ there exists $y \in N$ which belongs to ...
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What is the nice way of proving the convexity of ROC?

Let $Y_0$ and $Y_1$ be two continuous random variables on an interval of real numbers $[a,b]$, which follow two distinct probability measures $P_0$ and $P_1$, respectively. Let the probability ratio ...
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Fourier coefficient of convex function

On $I = [0, 2π]$ consider the function $f : I → \mathbb{R}$ to be convex. Define: $$a_k\pi := \int_0^{2\pi}f(x) \cos(kx)\,dx$$ Show that the convexity of $f$ implies that $a_k ≥ 0$ when $k ≥ 1$. ...
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Nearest point on Spherical Cap

Let $A \subset \mathbb{S}^n$ be a spherical cap. More specifically, there exists a point $v \in \mathbb{S}^n$ and $\epsilon > 0$ such that $A = \{u \in \mathbb{S}^{n}\mid v\cdot u \geq \epsilon\}$. ...
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Exercise 2 in Hatcher, section 1.2: the union of convex sets is simply connected

Here is my problem : I have convex subsets $X_1, \dots, X_n$ of $\mathbb R^m$ such that $X_i \cap X_j \cap X_k$ is never empty. I have to show that $\bigcup\limits_{i=1}^nX_i$ is simply ...
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weighted geometric mean

Ask for a hint to show following concave: $h(y) = y_1^{\theta1}...y_m^{\theta m}$ with $\theta_1+...+\theta_m=1$ and $\theta_i \geq 0$ If I do not want to use Hessians to show, any better way to ...
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About the strictly convexity of log-sum-exp function

The log-sum-exp function $f: \; \mathbb R^n \to \mathbb R$ is defined by $$f(x)=\ln \left (e^{x_1}+\cdots + e^{x_n} \right).$$ It is well-known that this function is convex, but I wonder that ...