# Questions tagged [high-dimensional-geometry]

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### Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

Imagine we have two size $m$ sets of points $X,Y\subset \mathbb{R}^n$. What is (time) complexity of testing if they differ only by rotation?: there exists rotation matrix $OO^T=O^TO=I$ such that $X=OY$...
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### Largest cell in an arrangement

Q. What is the complexity of finding the largest volume bounded cell in an arrangment of $n$ hyperplanes in dimension $d$? I feel I should know this... But I am not finding a definitive reference. ...
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### Computing volume of high-dimensional convex polyhedra

I am looking for software for computing/estimating volume of high-dimensional convex polyhedra. More specifically, I am interested in a program, which can handle bodies with $n$ vertices in $d$-...
335 views

### VC dimension of Voronoi cells in R^d?

Suppose I have $k$ points in $\mathbb{R}^d$. These induce a Voronoi diagram. If I assign to each of the $k$ points a $\pm$ label, these induce a binary function on $\mathbb{R}^d$. Question: what is ...
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### Far point queries in high dimensions

Given a set of points $X\subset R^d$ and a number $r\in R$, create a data structure for queries of the form: "given a point $q\in R^d$ return a point $x\in X$ with $\text{dist}(q,x)\ge r$". This is ...
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### Lattice generation inside d-dimensional unit ball

I am interested to know if there is a standard algorithm for generating all the lattice points inside the $d$-dimensional unit ball (with respect to the $\ell_2$-norm). The brute force approach is to ...
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### Random Projections and separability

I am new to machine learning and I am considering the following problem: Suppose you have clusters of points in $\mathbb{R}^N$ with $N$ large. The Johnson-Lindenstrauss lemma specifies how distances ...
Suppose we have two convex bodies $A$ and $B$, where $A \subseteq B$. Is it always true that $\mathrm{Vol}(A)/\mathrm{SurfaceArea}(A) \leq \mathrm{Vol}(B)/\mathrm{SurfaceArea}(B)$? It's true in all ...