Thesis (Selection of subject)Thesis (Selection of subject)(version: 368)
Thesis details
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Andersonova veta
Thesis title in thesis language (Slovak): Andersonova veta
Thesis title in Czech: Andersonova věta
Thesis title in English: Anderson's theorem
Key words: unimodalita|integrál|Brunn-Minkowského nerovnovst|konvexita|vícerozměrná míra
English key words: unimodality|integral|Brunn-Minkowski inequality|convexity|multivariate measure
Academic year of topic announcement: 2021/2022
Thesis type: Bachelor's thesis
Thesis language: slovenština
Department: Department of Probability and Mathematical Statistics (32-KPMS)
Supervisor: doc. Mgr. Stanislav Nagy, Ph.D.
Author: Bc. Filip Bočinec - assigned and confirmed by the Study Dept.
Date of registration: 29.09.2021
Date of assignment: 29.09.2021
Confirmed by Study dept. on: 18.11.2021
Date and time of defence: 07.09.2022 08:15
Date of electronic submission:06.05.2022
Date of submission of printed version:25.07.2022
Date of proceeded defence: 07.09.2022
Opponents: doc. RNDr. Petr Lachout, CSc.
 
 
 
Guidelines
Riešiteľ(ka) sa zoznámi so znením Andersonovej vety o integráloch cez konvexné množiny, a dokáže ju. Budú diskutované interpretácie, možné rozšírenia tejto vety, a jej aplikácie v pravdepodobnosti a štatistike.
References
T. W. Anderson (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc., 6:170–176.
S. Dharmadhikari and K. Joag-Dev (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Academic Press, Inc., Boston, MA.
A. P. Soms (1991). A note on equality in Anderson’s theorem. Comm. Statist. Theory Methods, 20(1):141–145.
M. D. Perlman (1990). T. W. Anderson's theorem on the integral of a symmetric unimodal function over a symmetric convex set and its applications in probability and statistics.
In The Collected Papers of T. W. Anderson, 1943-1985 (George P. H. Styan, ed.), 1627-1641. Wiley, New York.
Preliminary scope of work
Hustotu pravdepodobnostnej miery f na priestore R^d nazveme symetrická kvázikonkávna ak, i) f je symetrická okolo počiatku, t.j. f(x) = f(-x) pre každé x, a ii) všetky horné úrovňové množiny funkcie f definované ako {x v R^d : f(x) >= c} sú konvexné. Nech K je symetrická (t.j. K = -K) konvexná kompaktná množina v R^d s neprázdnym vnútrajškom. Klasická Andersonova veta hovorí, že zo všetkých posunutí (K + x) množiny K v priestore R^d, je integrál z f cez (K + x) maximálny ak x je počiatok 0.

Hlavným cieľom práce je spracovanie prehľadaného dôkazu tejto dôležitej vety, a diskusia o jej možných rozšíreniach a implikáciách. Za akých podmienok môžeme zaručiť, že x=0 je jediný posun množiny K ktorý maximalizuje integrál? Platí podobné tvrdenie bez predpokladu symetrie K a f? Je konvexita množiny K skutočne nutná? Niektoré z týchto otázok sú jednoduché, a odpovede na ne sú už dávno známe. Iné sa ukazujú ako omnoho zložitejšie, a sú iba veľmi málo preskúmané.
Preliminary scope of work in English
A density of a probability measure f on the space R^d is called quasi-concave if i) f is symmetric around the origin, i.e. f(x) = f(-x) for each x, and ii) all upper level sets of function f given by {x in R^d : f(x) >= c} are convex. Let K be a symmetric (i.e. K = -K) convex compact set in R^d with non-empty interior. The classical Anderson theorem says that, out of all shifts (K + x) of the set K in the space R^d, the integral of f over (K + x) is maximized if x is the origin 0.

The main goal of this thesis is to provide a self-contained proof of that important theorem, and a discussion on its possible extensions and implications. Under which condition can be guaranteed that x=0 is the only shift of the set K maximizing the integral? Does an analogous result hold true without the assumptions of symmetry of K and f? Is the convexity of K really needed? Some of these questions are simple and well explored. Others appear much more complicated, and are only very little studied.
 
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