All Questions
Tagged with convex-polytopes euclidean-geometry
20 questions
2
votes
0
answers
318
views
What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
- 43
5
votes
0
answers
167
views
Maximal number of vertices of the intersection of a flat and a hypercube
Consider the intersection of an $n$-dimensional hybercube and an $m$-dimensional flat (affine subspace) which contains the diagonal of the hypercube. This is a convex polytope. What is the maximal ...
- 1,368
2
votes
2
answers
294
views
Minimum Euclidean squared norm in the convex hull of points with rational coordinates
This is probably known, but I have not located a reference.
Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...
- 4,729
3
votes
1
answer
106
views
Solid angles at points in an orthosimplex
Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...
- 19.7k
4
votes
2
answers
494
views
addition theorems for hypersine
I learned from Wolfram MathWorld about hypersine, as being a dimensional analog trig function for hypersolid angles. There it is being defined by
The hypersine ($n$-dimensional sine function) is a ...
1
vote
0
answers
113
views
Two Questions on Tetrahedra and Platonic Solids [closed]
As was known to the ancients, two congruent regular tetrahedra can be inscribed in a cube and likewise 5 congruent regular tetrahedra can be inscribed in a regular dodecahedron. Is the converse to ...
- 157
2
votes
0
answers
87
views
Existence of a "generic enough" lattice point interior to a lattice triangle
Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
- 3,079
6
votes
1
answer
274
views
Example of convex n-gon that cannot be decomposed into k congruent convex polygons
I asked a related question here on MO without any answers yet.
The question is in the title - give an example of a convex $n$-gon that cannot be subdivided into $k>1$ congruent convex polygons.
...
- 15.8k
54
votes
5
answers
2k
views
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
- 10.1k
5
votes
1
answer
268
views
what's the formula of the inradius of a general simplex? [closed]
As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!
- 51
3
votes
1
answer
804
views
Approximation of a convex body by a contained polytope
This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...
- 591
4
votes
3
answers
499
views
Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?
For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation :
Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon $P_{...
- 4,757
15
votes
1
answer
640
views
Smallest regular simplex containing the unit cube in $R^n$
What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$?
In particular, is there $n$ such that $e_n<\sqrt{2}(n+1-\...
- 6,101
20
votes
1
answer
890
views
Tetrahedra passing through a hole
Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of edge-...
- 4,757
10
votes
3
answers
2k
views
On maximal regular polyhedra inscribed in a regular polyhedron
Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...
- 4,757
1
vote
1
answer
187
views
Straight Line Passing Through a Convex Region
Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, you can assume that ...
- 113
6
votes
0
answers
491
views
Minimum solid angle and aspect ratio of an $n$-simplex
In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.
In two ...
- 5,327
2
votes
1
answer
216
views
Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map
Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the ...
- 1,063
58
votes
14
answers
19k
views
Open problems in Euclidean geometry?
What are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit ...
6
votes
1
answer
715
views
Elementary problem about triangles inside a convex polygon
Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
- 1,318