All Questions
33 questions with no upvoted or accepted answers
20
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433
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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 ...
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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
0
answers
573
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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 ...
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7
votes
0
answers
316
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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 ...
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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, ...
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5
votes
0
answers
135
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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
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 ...
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3
votes
0
answers
136
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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 ...
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3
votes
0
answers
208
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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 ...
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3
votes
0
answers
76
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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 $...
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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 ...
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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
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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 ...
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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 ...
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2
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0
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71
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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 ...
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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^\...
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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
0
answers
42
views
On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
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1
vote
0
answers
59
views
What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
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1
vote
0
answers
44
views
On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
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1
vote
0
answers
40
views
Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
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1
vote
0
answers
52
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On families of lines that cut the boundary of a planar convex region in a specified ratio
We proceed from A claim on the concurrency of area bisectors of planar convex regions
This question is somewhat broad.
Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes ...
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1
vote
0
answers
38
views
Possible extensions of the perpendicular axes theorem for moment of inertia
This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...
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1
vote
0
answers
85
views
More on triangles inscribed in convex regions with one vertex fixed
We add a bit to On maximum perimeter triangles inscribed in convex regions with one vertex fixed. Let C be a convex planar region and P a point on its boundary.
Are there convex shapes C other than (...
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1
vote
0
answers
111
views
Maximizing the minimum curvature of a convex shape with a given volume in higher dimensions
Given any $d$-dimensional convex shape $S$ in the Euclidean space with $d\gg 1$, let $K_{\min}(S)$ be the minimum value of the Gaussian curvature of its boundary.
Question: What is the maximum value $...
- 1,677
1
vote
0
answers
76
views
Convex planar regions with optimal average 'centralness' and 'depth'
For a planar convex region $C$ and an interior point $P$ we define:
the centralness ratio at $P$ is
$$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\text{ is a ...
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1
vote
0
answers
153
views
Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
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1
vote
0
answers
57
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Shadows and planar sections of polyhedra – 2
This post continues Shadows and planar sections of polyhedra and On planar sections of 3D convex bodies
Shadows and planar sections of polyhedra gives an example demonstrating that shadows (orthogonal ...
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1
vote
0
answers
124
views
Number of lattice points in a structural symmetric convex body
Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space
\begin{equation}
\...
- 550
0
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0
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82
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On 'Bisecting sections' of 3D convex bodies
Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of ...
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0
votes
0
answers
49
views
Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?
By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.
Question: For specified ...
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0
votes
0
answers
74
views
The closest ellipse and circle to a given triangle - 2
We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...
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