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: Mgr. 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.