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Results tagged with mg.metric-geometry
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user 2906
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
3
votes
Number of independent distances between n points in d-dimensional Euclidean space?
The additional `constraints' missing
from these $N \choose 2$ interparticle distances are rank constraints
on the corresponding matrix $M$ of (squared) distances.
There is a matrix parameterizati …
- 8,336
1
vote
Variational characterization of curvature?
Take a circle of radius $r$ about a point $p$ (metric concept). Compute its
circumference (metric concept). Compare $C(r)$ to $2 \pi r$ in the limit
as $r \to 0$ to get the curvature $K(p)$ at $p$. …
- 8,336
2
votes
Shadow boundary on convex body in $\mathbb{R}^3$
This is not an answer but a shadow tangent Mohammad Ghomi has a wonderful paper
concerning a converse question: what are neccessary and sufficient conditions
on the shadows which insure the surfa …
- 8,336
4
votes
2
answers
151
views
Pushing a convex cone and equidistants
Let $K$ be a closed convex cone in an n-dimensional Euclidean space.
Suppose $K$ has non-empty interior. For $t > 0$
form the subcone $K_t$ consisting of all points in $K$ which lie a distance $ …
- 8,336
1
vote
Pushing a convex cone and equidistants
Conversations with friends today led to a solution.
A generic 4-sided convex polyhedral cone in 3-space led to a counterexample.
To be specific, suppose the 3-dimensional convex polyhedral cone
$K$ …
- 8,336
16
votes
Accepted
Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
The reason the problem is hard is that we do not have a good handle on what abnormal (=singular) geodesics can look like. See the chapter of my book that describes abnormal geodesics. Progress is …
- 8,336
20
votes
Nonconvex manhole covers
Having personally fallen halfway through a circular manhole in Somerville, MA,
I can say that this excuse for circular design is LOUSY. I stepped on
the metol disc and it flipped up,
a bit like a s …
- 8,336
9
votes
Probing a manifold with geodesics
Fix one $u$, its resulting geodesic probe, and vary $u$ slightly. Comparing the two geodesics
at the same arclength values yields a close approximation to the solution to the Jacobi equation al …
- 8,336
4
votes
4
answers
1k
views
Continuous pointwise ergodic theorem?
Let $\Phi$ be a homeomorphism of a compact metric space $M$
which preserves a regular Borel
probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. )
Under these hypothesis, I have two questio …
- 8,336
4
votes
Segments of Voronoi Diagrams on smooth manifolds. Are they geodesics?
More of a comment than an answer. Chapter 5 of William Goldman's
book `Complex Hyperbolic geometry' has a great detail of information on
the bisectors in complex hyperbolic space (so real dimension …
- 8,336
1
vote
Möbius transformation by 3 points in the Minkowski model
It is fun to see in an invariant way where the embedding
$\text{PGL}_2{\mathbb C}=\text{PSL}_2{\mathbb C}\hookrightarrow\text{PO}({\mathbb R};3,1)$
which Sasha speaks of `comes from'.
Take $ …
- 8,336
2
votes
Zeta function for curves in a manifold
In a slightly different spirit,
but still carrying the analogy between closed curves and primes,
The Selberg trace formula relates a sum over the lengths of
closed geodesics on a hyperbolic surfac …
- 8,336
4
votes
What's that shape? Inferring a 3D shape from random shadows
Joseph: this `answer' is perhaps more a comment than an answer, so apologies in advance. The deepest geometric work
I know of is this wonderful piece by Mohammad Ghomi. I quote from his
web page:
` …
- 8,336