Ph.D Thesis | |

Ph.D Student | Dmitriyuk Alon |
---|---|

Subject | Asymptotic and Probabilistic Estimates on Distortions of Linear Mappings between Convex Bodies |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Yehoram Gordon (Deceased) |

This thesis is about embeddings of subsets of the Euclidean space into Banach spaces using random mappings. Particular cases were considered by A. Dvoretzky and many other subsequent authors in the case of embedding Euclidean spaces of dimension n into general Banach spaces having higher dimensions.

To follow and expand on these ideas,
we consider a subset *T* of the *n*-dimensional Euclidean space and a
Banach space *Y*. Assume that a map *F* from *R*^{n} to *Y*
satisfies that for any *t*_{1}*, t*_{2} in *T* * A||
t*_{1}*- t*_{2}*||*_{2}*<||F(t*_{1}*)-F(t*_{2}*)||*_{Y}*<
B|| t*_{1}*- t*_{2}*||*_{2}.

Then we say *F* has distance
distortion at most *B/A* on *T*. Since in this thesis we consider
only linear mappings then by putting

*S={(t*_{1}*-t*_{2}*)/|| t*_{1}*-t*_{2}*|| _{2}):: t*

Hence we may consider subsets *S*
of the n-dimensional unit sphere *S*^{ n-1} and for *D>1*
we give conditions on Banach spaces *Y* for which there are linear maps *F*
from *R*^{n} to Y for which *Sup||F(s)||*_{Y}*/Inf||F(s)||*_{Y}*<D*
where the infimum and supremum are taken over all *s* in *S*.

The methods used in this thesis are probabilistic: We prove that random matrices give the required mappings under certain conditions. A particular case of general results proved in this thesis shows that when the target space is Euclidean we obtain the following result:

Let *r>0* and *0< k<
n* and let *{W*_{l}*}* be *p* affine subspaces of *R*^{n}
each of dimension at most *k*. Let *m=O(r ^{-2}(k?(p))* if

*||x-y||*_{2}*<||H(x)-H(y)||*_{2}*< _{ }(1)||x-y||*