# 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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### Efficiency of quasiconvex optimization

Summary: Could we minimize quasiconvex objectives in polynomial time? Whenever an objective function of an optimization problem can be formed as a convex function, this is considered as victory. ...
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### Proof or disproof of convexity for $f(x,y)=x^2y^2$

I'm trying to prove or disprove the convexity of $f(x,y)=x^2y^2$. This is part of a larger function but I think I proved that the rest of the function is convex using Hessian's. The other term in the ...
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### shrinking a convex hull around a set of polygons

I'm trying to find (Or design) an algorithm that will let me, after I have a convex hull, progressively shrink the hull towards the polygon set via increasing some parameter. I.e., if we use the ...
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### Subadditivity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Let $f(r) = \mu(A_r)^{1/n}$ where $\mu$ is ...
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### Convexity of a non linear optimization problem

I have a non linear optimization problem, namely: $$\min {\sqrt{(x-u)^2 + (y-v)^2 + (z-w)^2)}}$$ How can i show that the above function is convex. Doing via Hessian is a difficult task.
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### Is the function $\frac{f(x)}x$ increasing, if $f(x)$ is convex? [closed]

Suppose $x$ is real and positive. $f''(x) > 0$ (f is convex) is the function $h(x) = \frac{f(x)}x$ increasing? If so, is it strictly increasing? If yes, why? Thank you.
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### Continous family of $n$-gons

The statement of this problem asks to show that if $A$ and $B$ are two distinct convex $n$-gons there is a continous family of convex $n$-gons such that the first in that family is $A$ and the last is ...
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### True or false? “sum of an m-strongly convex and a convex function is m-strongly convex”

I would like to know if the following conjecture is true or false? If $f(x) = g(x) + h(x)$ where $g$ is m-strongly convex and $h$ is convex, then $f$ is m-strongly convex. NOTE: For a non-...
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### A connectivity-preserving function from a connected set onto an interval

Let $C$ be a connected set in the plane and $I$ the unit interval interval. Call a function $f$ from $C$ onto $I$ Connectivity-preserving if the following is true for every subset $I'\subseteq I$: ...
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### Prove that $A - B$ is closed if $A$ or $B$ is bounded (or both)

Suppose that $A$ and $B$ are two nonempty convex closed sets in $\mathbb{R}^n$, with $A \cap B = \emptyset$. Further, define $A - B = \{a - b \space | \space a\in A, b \in B\}$. Prove that $A - B$ is ...
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### When is $f(x,y,z)= \frac{x \cdot y}{z}$ convex?

I would like to know under what conditions $f(x,y,z)= \frac{x \cdot y}{z}$ is convex, pseudo-convex, or quasi-convex. I know that $g(x,y)= \frac{x^2}{y}$ $\text{when } y >0$ is convex and ...
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### Determining whether or not these spaces are convex

Consider $\{(x,y) \in \mathbb{R}^2: |x| +y^2\leq 5\}$ and $\{ (x,y) \in \mathbb{R}^2: y\geq x^2,y\leq e^{-x^2}\}$. Determine whether or not these two are convex sets. I have used the visual ...
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### Does every collection of edge vectors of a cone span a face of the cone?

Definition 1 : A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A ...
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### About a definition of “rank” of a matrix.

I am familiar with the definition of rank of a matrix as either (1) the maximal number of linearly independent rows or columns or (2) as the dimension of the image of the matrix. Another ...
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### Convex Analysis - How To Find Non Convex Set

I have a problem regarding the following exercise (I considered to put this question on mathematica.stackexchange, but I changed my mind and though this was the right place for this particular ...
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### Convex closure of the support of a Levy Process

Given a Levy Process $X_{t}$ on a filtered Probability Space $(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ with distribution-function $F_{t}$ for $X_{t}$. We look now for the cumulant transform $\phi_{1}$ ...
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### Unit ball with p norm in $\mathbb{R}^3$ space

I know unit ball for $p$-norm with $p = 2$ is a square, my confusion is how does it look like in $\mathbb{R}^3$ space. In $\mathbb{R}^3$ space it looks like a cuboid, is this correct ?
Show that $$f(\vec{x}) = \frac{1}{x_1 - \frac{1}{x_2 - \frac{1}{x_3 - \frac{1}{x_4}}}}$$ is convex when all denominators are greater than $0$.
Is $\displaystyle f(x_1,x_2) = x_1 - \frac{1}{x_2}$ a convex function? What if we restrict the values of this function to the positive reals?