All Questions
71 questions
88
votes
2
answers
7k
views
38
votes
7
answers
5k
views
Shortest path connecting two opposite points on a cube
Is it true, that a path connecting two opposite points (i.e. such that the segment joining them passes through the centre of mass of the cube) on the surface of the $d$-dimensional unit cube (with $d&...
- 1,431
21
votes
2
answers
1k
views
On convergence of convex bodies
Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...
- 21.8k
20
votes
0
answers
433
views
Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
- 13.6k
18
votes
3
answers
2k
views
Are the Platonic solids shadows of 4-polytopes?
Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection).
I am wondering if each of the five ...
- 151k
14
votes
4
answers
453
views
Smallest containing simplex
Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...
- 6,819
14
votes
0
answers
479
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?
After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
- 532
13
votes
1
answer
430
views
Detecting a hidden convex body with line probes
Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...
- 151k
13
votes
0
answers
573
views
What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
- 3,068
10
votes
3
answers
537
views
Perimeter-halving center of a convex shape
Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ perimeter-halving
if half the perimeter lies to one side of $c$
(and so half to ...
- 151k
9
votes
1
answer
370
views
Largest convex hull of a unit length path
What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
- 851
9
votes
2
answers
321
views
Is a polytope that has in-spheres for faces of all dimensions already regular?
Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points).
A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...
- 13.6k
7
votes
3
answers
377
views
Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
- 151k
7
votes
0
answers
316
views
Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
6
votes
4
answers
2k
views
Delaunay triangulations and convex hulls
This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
6
votes
1
answer
413
views
How many unit cubes are needed to 'hide' a unit cube fully in 3D?
Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
- 5,979
6
votes
2
answers
444
views
On planar sections of 3D convex bodies
Consider the space of planar sections of any given convex 3D body.
Basic Question: What is the lower bound for the ratio
$$\frac{\text{area of section of greatest perimeter}}
{\text{area of section of ...
- 5,979
5
votes
2
answers
241
views
On intersections of several convex regions
Question: Given n convex planar regions. Required to place them (in suitable position and orientation) so that that part of the plane lying under all the regions (their common intersection) is of ...
- 5,979
5
votes
2
answers
134
views
Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?
Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
- 13.6k
5
votes
0
answers
93
views
Which polytopes can be deformed while keeping their edge-lengths?
Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while
keeping its combinatorial type, and
keeping its ...
- 13.6k
5
votes
0
answers
139
views
On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
- 5,979
5
votes
0
answers
135
views
What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?
Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...
- 13.4k
4
votes
4
answers
536
views
Vertex-transitive polytopes in any dimension with any number of vertices?
Given positive integers $d$ and $v$ with $v \geq d+1$, does there always exist a (convex) vertex-transitive $d$-polytope with $v$ vertices? It seems that the answer should be "obviously" true, but I ...
- 51
4
votes
1
answer
333
views
n-simplex in an intersection of n balls
Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
- 195
4
votes
1
answer
203
views
Covering a convex body with its smaller homothetic copies
Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...
- 285
4
votes
1
answer
303
views
On maximum perimeter triangles inscribed in convex regions with one vertex fixed
Ref: Convex curves with many inscribed triangles maximizing perimeter
Given a planar convex region C. Let P be a variable point on its boundary.
Observations: When C is an ellipse, the variation in ...
- 5,979
4
votes
1
answer
124
views
Convex caps with prescribed edges and curvature
Let $G$ be the edge graph of a convex subdivision of a convex polygon $P$ in the plane. I would like to construct a convex polyhedral cap $C$ (with zero boundary values) over $P$ whose edges project ...
- 7,194
4
votes
0
answers
54
views
On ways to measure the difference between two planar convex regions
This earlier post attempted to quantify the difference between a pair of planar convex regions of equal area and perimeter using Hausdorff distance:
On comparing planar convex regions of equal ...
- 5,979
3
votes
1
answer
190
views
On some centers of convex regions based on partitions
These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'.
Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
- 5,979
3
votes
1
answer
205
views
How to show it is contained in a convex hull?
There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have
$$
\mathcal{H}(F_i)\...
- 43
3
votes
0
answers
136
views
If all max area planar sections of a solid are centrally symmetric, will the solid as whole be centrally symmetric?
It is known that every planar section of an ellipsoid is an ellipse - a centrally symmetric planar figure.
Are there convex solids other than ellipsoids with the property that all its planar sections ...
- 5,979
3
votes
0
answers
208
views
Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
3
votes
0
answers
76
views
A claim on planar sections of 3D convex bodies
Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $...
- 5,979
3
votes
0
answers
321
views
Polyhedrons and their centers of mass
Given a convex polyhedron, one considers 3 possibilities:
wireframe - only the edges of the polyhedron have mass which is uniformly distributed.
surface - only the surface is massive with uniform ...
- 5,979
3
votes
0
answers
52
views
Deformations that flatten small curvature
I'm trying to show that any 3-dimensional polyhedron with many vertices can be mildly deformed so that its vertices are no longer convexly independent. I suspect it suffices to look at a vertex with ...
- 31
2
votes
1
answer
302
views
Are two convex solids with all corresponding shadows equal in area congruent?
By shadow we mean the orthogonal projection of a convex 3D body P onto a 2D plane, for example, the shadow on the xy-plane, with P above (z>0) that plane and the light at L=(0,0,+∞). P an be freely ...
- 5,979
2
votes
1
answer
132
views
Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle
We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.
Consider a planar ...
- 5,979
2
votes
1
answer
84
views
'Constrained morphing' of planar convex regions
Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...
- 5,979
2
votes
1
answer
107
views
To find the convex planar region minimizing diameter when area and perimeter are given
The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...
- 5,979
2
votes
2
answers
164
views
Angle between a point in a convex polytope and the nearest point of a face
Let $P \subset \mathbb{R}^d$ be a convex polytope, and let $F$ be a face of $P$ (of co-dimension 1, let's say). Now let $x \in P \setminus F$ and let $y \in F$ be the nearest point of $F$ to $x$. Then ...
- 153
2
votes
1
answer
66
views
Optimal unions of planar convex regions
This post continues Optimal intersections between planar convex regions.
Question: Given two planar convex polygonal regions $C_1$ and $C_2$, how does one algorithmically find how to place and orient ...
- 5,979
2
votes
1
answer
426
views
Minkowski sum, zonotopes, convex hull
For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, ...
- 29
2
votes
1
answer
308
views
Intersection of the simplex with a linear subspace of codimension $2$
The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.
Let $S$ be the $n$-simplex:
$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\...
- 449
2
votes
1
answer
209
views
Cutting convex regions into equal diameter and equal least width pieces
The diameter of a convex region is the greatest distance between any pair of points in the region.
The least width of a 2D convex region can be defined as the least distance between any pair of ...
- 5,979
2
votes
0
answers
63
views
Convex planar regions such that every boundary point has a 'fair bisector' passing thru it
We add a little to On 'fair bisectors' of planar convex regions and A claim on the concurrency of area bisectors of planar convex regions .
A fair bisector of a planar convex region is a line ...
- 5,979
2
votes
0
answers
114
views
More on shadows of 3D convex bodies
Ref: Shadows and planar sections of polyhedra
By shadow we mean the orthogonal projection of a convex 3D body C onto a 2D plane, for example, the shadow on the xy-plane, with C above (z>0) that ...
- 5,979
2
votes
0
answers
71
views
On cutting convex regions with average values of quantities minimized
This post continues from Cutting convex regions into equal diameter and equal least width pieces - 2 and Cutting convex regions into equal diameter and equal least width pieces - 3
A basic (and to my ...
- 5,979
2
votes
0
answers
103
views
Polytopes with large dihedral angles
The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$.
The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
- 13.6k
2
votes
0
answers
415
views
Find the intersection between two convex hulls, in this specific case
We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$.
Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
- 233
1
vote
2
answers
1k
views
Is there always a parallelogram cross-section of parallelepiped contained in the smallest box
Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the ...