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Results tagged with discrete-geometry
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user 33089
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
9
votes
1
answer
158
views
Hyperplane arrangements whose regions all have the same shape
Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal transformation …
- 3,271
8
votes
1
answer
377
views
A "polar dual" for projective varieties?
Given a projective variety $X$ (over $\mathbb{C}$, say) with an affine paving $X=\sqcup_i C_i$, one can construct a poset $P_X$ on the set of cells $\{C_i\}$ by saying $C_i \leq C_j$ whenever $C_i \su …
- 2,537
6
votes
Accepted
How different can the constituents of an Ehrhart quasi-polynomial be?
Let $d=\dim(P)$. First, since $L(t,P)$ is non-decreasing in $t$, for any positive integer $n$ we have
$$f_i((n-1)D+i) \leq f_j((n-1)D+j) \leq f_i(nD+i) \leq f_j(nD+j)$$
whenever $i \leq j$. Thus it …
- 2,743