# Tagged Questions

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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### When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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### Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
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### What is the name of this property?

If there are 3 intervals, such that any 2 of them intersect, then all 3 of them intersect. For any 4 disks, if any 3 of them have a non empty intersection, then all 4 of them have a common ...
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### Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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### Distributed Convex Optimization Algorithm

Consider the convex optimization problem $$\min_{x_1, \cdots, x_N, y} \sum_{i=1}^{N} f_i(x_i,y)$$ $$\text{subject to: } x_i \in X_i \ \ \forall i, \ \ y \in Y, \ \ y = \sum_{i=1}^{N} x_i$$ ...
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### An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?

I would like to know an explicit method on constructing an everywhere discontinuous real function $F$ with the property: $$F((a+b)/2)\leq(F(a)+F(b))/2.$$ There is a non-constructive example (with the ...
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### Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
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### Second derivative positive $\implies$ convex

In proof of the following theorem; If $f$ has a second derivative that is non-negative (positive) over an interval then $f$ is convex (strictly convex). $f$ is in real number space., the book I ...
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### $\text{cl}(\text{int}(A)) = \text{cl}(A)$ when A is convex

How can one prove that $\text{cl}(\text{int}(A)) = \text{cl}(A)$, where $A \subseteq \mathbb{R}$ is convex? I know that if $A$ is convex, $\text{int(A)}$ and $\text{cl(A)}$ are convex too. I need ...
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### proving this inequality related to conjugate functions

For $x \in \mathbb{R}^n$ let us denote $x_{[i]}$ the $i$th largest component of $x$ s.t $$x_{[1]} \geq x_{[2]} \geq x_{[3]}\ge\cdots$$ The function $$f(x)= \sum_{i=1}^r x_{[i]}$$ is the sum of $r$...
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### How should I prove a set is convex?

Given a set $$\mathbf{S} = \{ \mathbf{x}\:|\: \mathbf{x}^T\mathbf{V}\mathbf{x}=1 \}$$ where $\mathbf{V}$ is a positive semidefinite matrix. How to prove this set is convex? I tried in the ...
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### Is $f(x)x$ convex for increasing function $f$?

Suppose that $f:(0,\infty)\to[0,1]$ is strictly increasing and infinitely differentiable. I have an intuition that $$g(x)=f(x)x$$ should be convex in $x$ (i.e. increasing at accelerating rate) ...
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### How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
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Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le \... 1answer 446 views ### Support function of a convex domain Let$Q$be a compact, convex domain in the plane, with smooth boundary$\partial Q$. We further assume that the origin is contained in$Q$. For a concrete example, let's take an ellipse$\frac{x^2}{a^...
I was studying the Stephen Boyd's textbook on convex optimization and have a question. The book says the following: The singular values of $A$, $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n$ are ...