Problems from Geometrical Functional Analysis

Doctoral Dissertation uoadl:2928553 84 Read counter

Unit:
Department of Mathematics
Library of the School of Science
Deposit date:
2020-11-17
Year:
2020
Author:
Skarmogiannis Nikolaos
Dissertation committee:
Γιαννόπουλος Απόστολος (Επιβλέπων)
Καθηγητής, τμήμα Μαθηματικών, Ε.Κ.Π.Α.

Γατζούρας Δημήτριος (Συμβουλευτική και Εξεταστική επιτροπή)
Καθηγητής,τμήμα Μαθηματικών,Ε.Κ.Π.Α.

Μερκουράκης Σοφοκλής (Συμβουλευτική και Εξεταστική επιτροπή)
Καθηγητής,τμήμα Μαθηματικών,Ε.Κ.Π.Α.

Δοδός-Ντοντός Παντελεήμων (Εξεταστική επιτροπή)
Αναπληρωτής καθηγητής, τμήμα Μαθηματικών, Ε.Κ.Π.Α.


Κανελλόπουλος Βασίλειος (Εξεταστική επιτροπή)
Αναπληρωτής Καθηγητής, Σ.Ε.Μ.Φ.Ε., Ε.Μ.Π.

Τσολομύτης Αντώνιος (Εξεταστική επιτροπή)
Καθηγητής,τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου

Χελιώτης Δημήτριος (Εξεταστική επιτροπή)
Αναπληρωτής καθηγητής, τμήμα Μαθηματικών, Ε.Κ.Π.Α.
Original Title:
Προβλήματα Γεωμετρικής Συναρτησιακής Ανάλυσης
Languages:
Greek
Translated title:
Problems from Geometrical Functional Analysis
Summary:
Αγγλικά ή άλλη γλώσσα:
We study a number of questions from Geometric Functional Analysis using
geometric, analytic and probabilistic methods.

1. Vector balancing problems. We obtain an improved version of a result of
D. Hajela concerning a well-known problem of Komlos. We also generalize
this result to independent random vectors that are uniformly distributed
in the Euclidean unit ball or an arbitrary symmetric convex body, and any
norm.

2. Weighted sums of log-concave random vectors. We obtain upper bounds for
the expected value of the norm of a weighted sum of independent random
vectors distributed according to a log-concave isotropic probability
measure, answering a question of V. Milman. We also present applications
to "randomized" versions of vector balancing problems.

3. Random convex sets. We study two classes of random convex sets and
provide upper and lower bounds for the expected value of their volume.

4. Affine quermassintegrals of random polytopes. We confirm, from an
asymptotic point of view, a related conjecture of Lutwak for some broad
classes of random polytopes.

5. The symmetric average and the MM*-inequality for isotropic convex
bodies. We discuss two known problems from asymptotic convex geometry. The
first problem asks for estimates on the symmetric average sav(K) of a
convex body K, while the second one concerns upper bounds for the mean
width and the mean norm of an isotropic convex body. We provide simpler
proofs for the best, up to now, known results.
Main subject category:
Science
Keywords:
Log-concave probability measures, Convex bodies, Isotropic constant, Random polytopes, Vector balancing problems, Komlos conjecture
Index:
No
Number of index pages:
0
Contains images:
No
Number of references:
115
Number of pages:
134
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