All Questions
16 questions
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The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
- 5,979
8
votes
2
answers
489
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Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,149
2
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1
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190
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Estimating the volume of a convex shape in higher dimensions based only on normal sections
We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ ...
- 1,677
1
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111
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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
6
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1
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388
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Covering number estimates on closed Riemannian manifolds
Let $(M^n,g)$ be an $n$-dimensional compact and connected Riemannian manifold with sectional curvature bounded above and below by $c,C$. Is it possible/known how to express the external covering ...
- 195
1
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64
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Euclidean embedding of the mesh
$M$ is a topological mesh, i.e. triple $M=(V,E,F)$, where $V$ is the vertex, $E$ is the edge and $F$ is the face, such that $M$ is homeomorphic to the sphere.
Suppose that we have a metric $l :E\...
- 1,915
14
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1
answer
642
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Which convex bodies can be captured in a knot?
Which convex bodies can be captured in a knot?
This question is based on the discussion in "Is it possible to capture a sphere in a knot?".
We assume that the knot is made from an ...
- 45k
6
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3
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365
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Sliding through a curvature-bounded tube: Maximum volume?
My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.
Q1. Is it the case that the maximum convex volume body inside a ...
- 151k
17
votes
5
answers
883
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Rigidity of convex polyhedrons in $\mathbb R^3$ with faces removed
Take a convex polyhedron $P$ in $\mathbb R^3$ and remove all the faces, i.e. leave only the edges. Call this graph $E$. Let us now try to continuously deform $E$ in $\mathbb R^3$ so that all the edges ...
- 14.3k
4
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1
answer
124
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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 ...
- 7,149
6
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191
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Cut locus on a hypercube
Inspired by the question, "Shortest path connecting two opposite points on a cube":
Q. What does the cut locus with respect to one corner of a hypercube
in $\mathbb{R}^d$ look like?
"The cut ...
- 151k
2
votes
1
answer
118
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Characterization of the medial axis of a surface
I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...
- 163
3
votes
1
answer
292
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Existence of Simple Closed Straightest Geodesics
There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...
- 133
6
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3
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482
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Herringbone partitions of regions and surfaces
Let $R \subset \mathbb{R}^2$ be a region of the plane bounded
by a Jordan curve. The boundary $\partial R$ could be a polygon,
or a smooth curve—there are variations depending upon boundary ...
- 151k
1
vote
0
answers
371
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Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
1
vote
2
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1k
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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 ...