Ph.D Thesis | |

Ph.D Student | Asinowski Andrei |
---|---|

Subject | Geometric Permutations in the Plane and in Euclidean Spaces of Higher Dimension |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Meir Katchalski |

Let *A*={*A*_{1},
*A*_{2}, …, *A _{n}*} be a family of disjoint convex
sets in Euclidean space

It
is known that a family of any number of disjoint translates of a convex set in
the plane **R**^{2} has no more than 3 geometric permutations. It
was conjectured that constant bounds exist in higher dimensions. We give a
construction in **R**^{3} that refutes this conjecture: a family of
2*n *disjoint translates of a convex set in **R**^{3}, with *n+*1
geometric permutations.

It
is known that certain collections of permutations cannot be geometric
permutations of the same planar family. We extend this observation to higher
dimensions, proving that: (1) for each natural *k*, each family of *k*
permutations is realizable in **R**^{2k-1}_{ }(and
therefore in higher dimensions); (2) for each natural *k*, there is a
family of *k* permutations which is non-realizable (forbidden) in **R**^{2k-2}_{
}(and therefore in lower dimensions as well).

Finally, we study realizable
and forbidden families of geometric permutations in **R**^{2}. We
prove two necessary and sufficient conditions for realizability of a family {*p*,
*q*} of two permutations as geometric permutations of a planar family. Regarding families of more than two permutations, we
give two necessary conditions for realizability in **R**^{2}, and show
by an example that they are not sufficient.