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Results tagged with convex-polytopes
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user 20595
Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
6
votes
Some polytopes in $\mathbb R^n$ whose vertices have coordinates 1, -1 or 0
View the two examples, I think $P(n,k)$ is the $(n-k)$-rectified $n$-hypercube or the $(k-1)$-rectified $n$-cross-polytope (same thing). I believe the notion of rectification will be very helpful for …
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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 …
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3
votes
Basic question about polytope duals
While the answer to the question of OP is "no", I would like to mention a question raised by Grünbaum and Shephard in "Some problems on polyhedra.", J. Geom. 29 (1987), no. 2, 182–190, for the informa …
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3
votes
Polytope with indegree-increasing property.
This answer benefited from a discussion with I. Izmestiev.
Consider a $3$-dimensional simple polytope $P$. The indegrees of the vertices can only be $0$, $1$, $2$ or $3$. The lowest vertex has inde …
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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 …
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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 …
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