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Results tagged with convex-polytopes
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user 38468
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
9
votes
Basic question about polytope duals
This is false, even in dimension $3$. Take a regular icosahedron. Then wiggle the vertices a bit -- still get an icosahedron, but it is not regular. If you take the barycenters and the convex hull, yo …
- 5,186
6
votes
Accepted
Estimates on the number of vertices of reflexive polytopes
A "cube" $[-1,1]^n$ has $2^n$ vertices and is reflexive.
- 5,186
3
votes
When are Ehrhart functions of compact convex sets polynomials?
I believe that the strong form of the conjecture is false. In lieu of a simple counterexample, let me point you towards a centrally symmetric 10-gon $\hat P$ in arXiv:0801.2812, Figure 6. It is a bit …
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2
votes
polynomial expression for counting number of integral points of a set
Let's see what happens in dim 2. You have $conv((0,0),(ra_1+sb_1,0),(0,ra_2+sb_2))$. The number of points in the closed triandle $(0,0),(A,0),(0,B)$ is $(A+1)(B+1)/2$ plus half the number of points on …
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2
votes
Lattice points in dilated polytopes and sumsets
In general, $g(n)$ is strictly less than $f(n)$, although they have the same leading term.
One sufficient condition to get $g(n)=f(n)$ is to require smoothness of the corresponding toric variety. Comb …
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0
votes
Lattice points in dilated polytopes and sumsets
Since you have changed the formulation of the question slightly to now require identity for all $n$, not just for large ones (or maybe it was just me misreading the original post),
let me offer a suff …
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