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Results tagged with open-problems
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user 68969
If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"
12
votes
Not especially famous, long-open problems which anyone can understand
A list of fundamental geometry problems with simple intuitive statements involving curves and surfaces in Euclidean space is maintained at:
Open Problems in Geometry of Curves and Surfaces
Five of t …
13
votes
Shortest closed curve to inspect a sphere
The baseball stitches curve suggested by Gjergji Zaimi appears in another paper of Zalgaller:
V. A. Zalgaller. Extremal problems on the convex hull of a space curve. Algebra i Analiz, 8(3):1–13, 1996 …
- 7,179
21
votes
Shortest closed curve to inspect a sphere
I have recently finished a paper called
The length, width, and inradius of space curves
where it is shown that the length $L$ of any closed curve $\gamma\colon[a,b]\to \mathbf{R}^3$ inspecting the u …
- 7,179
1
vote
0
answers
126
views
What is an umbilic point of a convex polyhedron?
An umbilic point of a smooth ($C^2$) convex surface in Euclidean 3-space is a point where the principal curvatures are equal. Is there some good way to generalize this notion to convex polyhedra? See …
- 7,179
13
votes
2
answers
865
views
Intrinsic vs Extrinsic geometry of convex surfaces
By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is …
- 7,179
22
votes
3
answers
1k
views
Equilaterally triangulated surfaces with prescribed boundary
There is a problem in Richard Kenyon's list (Wayback Machine) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest pr …
- 7,179
80
votes
1
answer
3k
views
Converse to Euclid's fifth postulate
There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, o …
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3
votes
Shortest closed curve to inspect a sphere
In another paper with James Wenk, we have shown that the condition in Zalgaller's conjecture that the curve lie outside the sphere is not necessary, that is, the inequality $L\geq 4\pi$ holds for all …
- 7,179
10
votes
0
answers
263
views
Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly p …
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44
votes
Accepted
Shortest closed curve to inspect a sphere
James Wenk and I just finished a paper proving Zalgaller's sphere inspection conjecture for closed curves:
Shortest closed curve to inspect a sphere.
We show that in $R^3$ any closed curve $\gamma$ w …
- 7,179
38
votes
0
answers
1k
views
Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of p …
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