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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
5
votes
Accepted
Stable equilibria of points on the 2-sphere
This is the famous Thomson problem. You can find a list of optimal configurations and many references on the Wikipedia page. Your intuitions for $n=7, 8, 9, 20$ are wrong, and $n=5$ is not that obvi …
5
votes
Three-dimensional Apollonian spirals
According to the discussion in Coxeter (1968), the tangent points lie asymptotically on a concho-spiral, so the distribution is not uniform on the sphere, but is uniform on a circle.
By the way, the …
3
votes
Is there a midsphere theorem for 4-polytopes?
I recently showed that:
The graph of a stacked $4$-polytope is $3$-ball packable if and only if it does not contain six $4$-cliques sharing a $3$-clique.
While Eppstein, Kuperberg and Ziegler 20 …
1
vote
Convex caps with prescribed edges and curvature
Given Gaussian curvatures at the vertices, there is a unique lift that realizes these curvatures, as you can see from Igor's note.
Given a graph, the set of liftings that projects to this graph form …
5
votes
Is there a midsphere theorem for 4-polytopes?
In a recent paper of Padrol and me, we studied several generalizations of this problem. http://arxiv.org/pdf/1508.03537v1.pdf
Regarding Q1, Yoav already mentioned Schulte's work, and Gil mentioned t …