1,578 research outputs found
LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies
We extend the classical LR characterization of chirotopes of finite planar
families of points to chirotopes of finite planar families of pairwise disjoint
convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a
chirotope of finite planar families of pairwise disjoint convex bodies if and
only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the
set of 3-subsets of J is a chirotope of finite planar families of pairwise
disjoint convex bodies. Our main tool is the polarity map, i.e., the map that
assigns to a convex body the set of lines missing its interior, from which we
derive the key notion of arrangements of double pseudolines, introduced for the
first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio
Novel Split-Based Approaches to Computing Phylogenetic Diversity and Planar Split Networks
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Realization spaces of arrangements of convex bodies
We introduce combinatorial types of arrangements of convex bodies, extending
order types of point sets to arrangements of convex bodies, and study their
realization spaces. Our main results witness a trade-off between the
combinatorial complexity of the bodies and the topological complexity of their
realization space. First, we show that every combinatorial type is realizable
and its realization space is contractible under mild assumptions. Second, we
prove a universality theorem that says the restriction of the realization space
to arrangements polygons with a bounded number of vertices can have the
homotopy type of any primary semialgebraic set
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