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master's thesis
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Convex Sets
Petar Nujić (2017)
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Pure Mathematics
Algebra and Calculus Research Group
Metadata
TitleKonveksni skupovi
AuthorPetar Nujić
Mentor(s)Mihaela Ribičić Penava (thesis advisor)
Abstract
Na početku ovog rada uvodimo definiciju afinog skupa i njegovu geometrijsku interpretaciju. Povezujemo pojam vektorskog potprostora sa afinim skupovima i dokazujemo tvrdnje koje vrijede za njih, te definiramo afinu ljusku. Zatim se upoznajemo s pojmom konveksnih skupova. Također spominjemo konveksnu ljusku pomoću koje možemo, od bilo kojeg skupa, načiniti konveksan skup. Iskazujemo i dokazujemo osnovne teoreme konveksne ljuske. Obrađujemo operacije koje čuvaju konveksnost i algebru s konveksnim skupovima. Definiramo konveksne funkcije pomoću epigrafa. Uvodimo pojmove efektivne domene i iskazujemo jedan bitan teorem pomoću kojeg pokazujemo konveksnost funkcija. Nakon toga je navedena karakterizacija konveksnih funkcija pomoću Jensenove nejednakosti i derivacija. Zatim, iskazujemo i dokazujemo Carathéodoryjev teorem. Osim afinih i konveksnih skupova upoznajemo se s (konveksnim) konusima, navodimo važna svojstva koja vrijede za njih i definiramo konusnu ljusku. Na kraju rada povezujemo konveksnost s topologijom.
Keywordsaffine sets affine hull convex sets convex hull convex functions convex cones conical hull Carathéodory’s theorem
Parallel title (English)Convex Sets
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
Lower level organizational unitsChair of Pure Mathematics
Algebra and Calculus Research Group
PlaceOsijek
StateCroatia
Scientific field, discipline, subdisciplineNATURAL SCIENCES
Mathematics
Mathematical Analysis
Study programme typeuniversity
Study levelintegrated undergraduate and graduate
Study programmeMathematics and Computer Science
Academic title abbreviationmag. educ. math. et inf.
Genremaster's thesis
Language Croatian
Defense date2017-12
Parallel abstract (English)
At the beginning of this paper we introduce the definition of affine sets and their geometrical interpretation. We connect the terms of vector subspaces to affine sets and offer some proofs which apply to them, and we define the affine hull. Then we introduce the definition of convex sets. Furthermore, we introduce the convex hull out of what any convex set can be built. We process operations that preserve convexity and the algebra of convex sets. We define the convex function through the epigraph. Furthermore, we provide the terms of effective domain and indicate one important theorem which helps us showing the convexity of a function. After that, we give the characterization of convex functions with the Jensen inequality and derivative. Then we mention the Carathéodory theorem and offer the proof. Apart from affine and convex sets we define (convex) cones and mention their properties and define their conical hull. At the end we connect convexity with topology.
Parallel keywords (Croatian)afini skupovi afina ljuska konveksni skupovi konveksna ljuska konveksne funkcije konveksni konusi konusna ljuska Carathéodoryjev teorem
Resource typetext
Access conditionOpen access
Terms of usehttp://rightsstatements.org/vocab/InC/1.0/
URN:NBNhttps://urn.nsk.hr/urn:nbn:hr:126:403773