Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Author:
Παπανικολάου Σουλτάνα
Supervisors info:
Γιαννόπουλος Απόστολος Καθηγητής (επιβλέπων), Κατάβολος Αριστείδης Καθηγητής, Μερκουράκης Σοφοκλής Καθηγητής
Original Title:
Βέλτιστη μεταφορά του μέτρου και γεωμετρικές ανισότητες
Translated title:
Optimal transportation of measure and geometric inequalities
Summary:
The mass transportation problem was initially introduced by the French geometer
Monge and it was solved long later under speci fic assumptions on probability
measures
and cost function. At this paper we mention the basic tools that were used and
led to the
solution of this problem, major of which are the Kantorovich duality theorem as
well as
MacCann's theorem, which compose a revised version of Brenier's theorem and it
solves
the initial Monge's problem via geometric arguments.
Kantorovich invented a comfortable notion of the distance between probability
mea-
sures as follows: the distance between two probability measures have to be the
optimal
transportation cost from one to the other, if we assume that cost is the
distance function
between those two measures. This distance of probability measures is known
today as the
Kantorovich-Rubinstein distance. One more useful tool which results from this
theory
is the Wassertein distances, which are distances that depend on the value of
the optimal
cost function, when cost is a power of the distance. In the last part of this
paper are
presented the applications of this theory on some basic and useful geometric
inequalities,
which we prove afresh via optimal transportation of measure.
Keywords:
Optimal transportation of measure, Kantorovich duality theorem, Geometric inequalities, Wasserstein distances, McCann's theorem
