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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Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?
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References for the following functional

In many of the types of problems Ive looked at the following quantity keeps arising and I was wondering if anyone knew any references I could look at to learn some its properties. Take any function ...
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$A$ be a non-empty closed convex subset of a Hilbert space $H$ , is the distance from $A$ always attained at a unique point in $A$ ?

Let $A$ be a non-empty closed convex subset of a Hilbert space $H$ , then is it true that for every $b \in H$ , $\exists$ unique $x_b \in A$ such that $||x_b-b||=d(b,A)=\inf \{||b-x||:x \in A\}$ ?
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Why is this function, related to SVM derivation, non-convex?

I'm working through a support vector machines tutorial. In eventually deriving the solvable objective function, the following objective function (to be maximized) was proposed, but dismissed as ...
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28 views

How to prove that the following function is convex

I have a following function that I would like to show it is convex in $x$, however the function is not a sum of convex functions so I think I need to specify some conditions. However I don't know how ...
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26 views

Is convex or non convex function?$J(u,c)=\int K(x).u.(f(x)-c)^2dx$

I have a function such as $$J(u,c)=\int K(x).u.(f(x)-c)^2dx$$ where $f(x):\Omega \to R$; c is constant; $0 \le u \le 1$; and K(.) is gaussian kernel. My question is that : Is J convex or non-convex ...
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How to understand a proposition of subgradient

The question is from the following: Convex Optimization Algorithm (p.512)----- Prof. Bertsekas Let $f: R^n \rightarrow (-\infty, \infty]$ be a proper convex function. For every $x \in ...
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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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Find parameters of a quadratic surface given 3 points

I have 3 points in the space each defined as a vector with its two coordinates $\eta_k=(x~~ y)^T$. Given $\eta_1,~ \eta_2$ and $\eta_3$ I would like to find the parameters $Q,~ P$ and $b$ of the ...
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+50

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
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Volume of Convex Polytope with rational Entries

I have the following question: In this article Polytope volume computation it is stated that when considering a bounded convex polytope $P=\{x \mid Ax\le b\}$ with the matrix $A$ and the vector $b$ ...
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47 views

Calculating the distance function and projection

Let $\Omega=\{x\in\mathbb R^n\mid\langle a,x\rangle=b\}$. We define the distance function and projection as follows $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
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21 views

Supporting lines of closed Jordan curve

Given a simple closed Jordan (i.e. continuous) curve $\gamma:[a,b]\to\mathbb{R}^2,\ \gamma([a,b])=C_\gamma$, how can I prove that $D_\gamma$ (the set of all interior points of $\gamma$ toghether with ...
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1answer
52 views

Convex function inequality for Euclidean norm: $\|(f(x_1),\cdots,f(x_n))\|_2\leq f(\|x\|_2)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive, convex, continuous function such that $f(0)=0$. (If you wish you can also suppose $f$ to be monotone increasing.) I would like to prove or to ...
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32 views

Difference of support functions and its minimum points

Let $A$ and $B$ are convex, compact sets in $\mathbb{R}^n$. We have known that $$\max_{a\in A}\min_{b \in B} \|a-b\|=\sup_{\|g\|\le1}(\sigma_A(g)-\sigma_B(g)),$$ where $\sigma_M(x)=\sup_{u\in ...
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Strongly convex, bounded from below by a quadratic function.

A strongly convex function $V: \mathbb{R}^d \rightarrow \mathbb{R}$ with negative parameter is given, i.e. $$ V(tx + (1-t)y) \leq tV(x) + (1-t) V(y) - \lambda t(1-t) | x -y |^2 , $$ with ...
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Simple question on convexity [on hold]

If $\epsilon,\lambda\in(0,1)$, $a,b\in(1-\epsilon,1+\epsilon)$, then is $$\lambda a+(1-\lambda)b\in(1-\epsilon,1+\epsilon)?$$
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58 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$, . Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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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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1answer
19 views

Interior of difference of two convex sets

Let $A, B$ be two nonempty convex set in normed space $X$. We always have $$ \text{int}(A)\bigcap B\ne\emptyset\;\Longrightarrow\; 0\in\text{int}(A-B). $$ Indeed, suppose that $\text{int}(A)\bigcap ...
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Is it convex function?

I have a function and I don't know it is whether convex or non-convex: $$J(c,\alpha)=\int_\Omega ( \alpha c-I(x))^2u \, dx+ \|\alpha\|^2$$ where $0 \le u \le 1$, $I(x): \Omega \to R$, $c$ is constant ...
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If $A\subset B$, then $\text{ri}\, A\subset\text{ri}\,B$?

Is it true that If $A\subset B$, then $\text{ri}\, A\subset\text{ri}\,B$? Let $u\in\text{ri}\,A$, then there is $\epsilon>0$ such that $$\mathbb B(u;\epsilon)\cap\text{aff}\,A\subset A\subset B$$ ...
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50 views

Lower semi-continuity of particular function

Let $F : \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a set-valued map, locally bounded, upper semi-continuous, and taking nonempty, convex and compact values. Let $f : \mathbb{R}^n \to \mathbb{R}$ ...
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proving that $(\text{aff}\,C-\text{aff}\,C)\subset\text{aff}(C-C)$

In proof of Theorem 6.4.1, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that $\epsilon^{-1}(C-\text{rge}\,A)\subset\text{aff}\,(C-C)$, that I can't verify ...
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18 views

Curvature of strictly concave function: scaling of derivative

I am looking at a strictly concave function $f: R_+ \to {R}$ with $f' >0$ and $f'' < 0$. I require the following property which I don't fully understand: $$ k f'(kx) \geq f'(x) \quad \forall ...
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Supporting hyperplane of a convex set

Let $\Omega$ be a bounded convex set in $\mathbb{R}^n$, and let $\partial \Omega$ denote its boundary. Fix a point $p$ in $\Omega$, and let $c$ denote the point on $\partial \Omega$ that is closest ...
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any good way to approximate this non-convex function with convex function?

There is a non-convex constraint in my optimization problem, which is given by $\displaystyle -xy\log\left(1+\frac{z}{xy}\right)$. Obviously, it is neither convex or concave. Is there any good convex ...
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353 views

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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Equivalence of two properties [closed]

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and $\alpha>0$. Prove that the two following properties are equivalent: (1) For all $|\delta|\leq\alpha$, $\lambda\in (0,1)$, and $x_0,x_1\in\mathbb{R}^n$ ...
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proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

In Proposition 6.4.1 we want to prove that if $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone, then $\text{cl rge}\,A$ and $\text{ri rge}\,A$ are convex. In proof we arrive to the ...
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Existence of a continuously differentiable function from the projection of a convex set

Let $V$ be an open convex set of $\mathbb{R}^{n+1}$ and let $U$ be the projection of $V$ onto $\mathbb{R}^n$ - i.e. the set of $x \in \mathbb{R}^n$ such that there is some $y \in \mathbb{R}$ with ...
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$f$ convex: exists linear map $g$ s.t. $f\geq g$?

Let $f:\mathbb R\to\mathbb R$ be convex. Does there exist a linear map $g(y)=ay+b$ such that $f(y)\geq g(y)$ and $f(x)=g(x)$? Clearly, if f is differentiable we can argue with the tangent line, but ...
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$x=argmin_{x\in A}||y-x||_2$ iff $\langle y-x,z-x\rangle \leq0$ for all $z\in A$

Consider $x\in A\subset\mathbb{R^n}$ with A closed and convex. How can you see that $$x=argmin_{x\in A}||y-x||_2$$ iff $$\langle y-x,z-x\rangle \leq0$$ for all $z\in A$. I tried using ...
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Interior of closed ball

I'm an absolute beginner in Convex analysis. I'm wondering how the following statement is true. I just got this from a lecture notes and unfortunately no proof is provided. "The interior of the ...
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Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$

Assume we have have $3$ points $x_1, x_2$ and $y$ and $||x_1-y||=||x_2-y||$. How do we prove that the distance between $y$ and the convex combination of $x_1$ and $x_2$ is smaller than that between ...
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1answer
25 views

Interpreting a condition about CDF

Let F(X) be a strictly increasing CDF which admits a positive density f(x). Is the condition x/F(x) being non-increasing (aka, weakly decreasing) equivalent to saying that F(x) is convex? If not, what ...
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Is there any way to make the following function convex?

I need to find optimal lagrangian multiplier vectors for a quadratic programming problem subject to three quadratic equality constraints and several other linear inequality constraints. I would like ...
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Proving $x^4$ is strictly convex

I'm not sure how to prove $f(x) = x^4$ is strictly convex using just the definition of strict convexity: $$f((1-t)x+ty) < (1-t)f(x)+tf(y)$$ for $0<t<1$. Is this just an algebra slog? If so, I ...
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Is $f(x) = \frac{(x+\alpha)^3}{(x+\beta)(x+\gamma)}$ quasiconvex?

Is the function \begin{equation} f(x) = \frac{(x+\alpha)^3}{(x+\beta)(x+\gamma)} \end{equation} where $0 \leq \beta \leq \alpha$ and $0 \leq \gamma\leq \alpha$ quasiconvex? $x$ can be assumed to ...
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Affine, surjective map between convex sets

The setup for my question is the following: I have a compact and convex subset $K$ of some locally convex topological vector space. Within $K$ there is a $T\subset K$ which is compact and convex and ...
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Determining the sign of a term

I have a problem in proving the sign of a term. It is as follows: $$x=\dfrac{1-a}{b_1b_2-a}+1,\qquad y=\dfrac{1-a}{b_1-a}+\dfrac{1-a}{b_2-a},\qquad z=x-y$$ with $0<b_1<1,\quad ...
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1answer
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Proof of relationship of subgradients of function to convexity of function

I am trying to follow the proof of the first claim of Proposition 7 on this page: https://blogs.princeton.edu/imabandit/2013/02/05/orf523-advanced-optimization-introduction/ Basically, we are given: ...
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How to prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++ [duplicate]

How can i prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++
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convexity proof of a function including ln and sums

$$f(x_1,\dots,x_n)=\sum\limits_{i=1}^nx_i\ln x_i-\left(\sum\limits_{i=1}^nx_i\right)\ln\left(\sum\limits_{i=1}^nx_i\right)\rightarrow R_{++}^n$$ How can I prove this is convex on $R_{++}^n$? I tried ...
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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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contractivity to prove that an entire function is of finite exponential type?

Suppose that $G(z)$ is an entire function. If I can show that G is contractive within some compact and convex subset of $\mathbb C$, is it easier from that point to establish that $G$ is of finite ...
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1answer
35 views

Theorem 3.1 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: Is the notion of convex set valid in complex vector spaces?

Is the notion of convex sets valid and meaningful for complex vector spaces? Or, do we need to restrict ourselves to real vector spaces and normed spaces when we talk about convex sets? The ...
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1answer
30 views

Approximate model of a convex/concave surface

I have a set of measurements in 3d that yields a concave surface of a function $f(x,y)$ that I don't know its expression. I am thinking to approximate the function to a model where any point from the ...
2
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2answers
32 views

Prove local minimum of a convex function is a global minumum (using only convexity)

I'm studying for a calculus exam, and have come across this question in the textbook which I have problem solving; Let $C\subseteq \mathbb{R}^d$ a convex set, and let $f:C\rightarrow \mathbb{R}$ ...
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1answer
19 views

Strong convexity and strong smoothness duality

A function $f$ is said to be strongly convex with respect to a norm $\|\cdot\|$ at a point $y$ if $f(x) \geq f(y) + \nabla f(y)^T(x-y) + \frac{1}{2}\|x-y\|^2.$ It is said to be strongly smooth with ...