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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
Cite this document

Nujić, P. (2017). Konveksni skupovi (Master's thesis). Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:403773

Nujić, Petar. "Konveksni skupovi." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2017. https://urn.nsk.hr/urn:nbn:hr:126:403773

Nujić, Petar. "Konveksni skupovi." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2017. https://urn.nsk.hr/urn:nbn:hr:126:403773

Nujić, P. (2017). 'Konveksni skupovi', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 15 November 2019, https://urn.nsk.hr/urn:nbn:hr:126:403773

Nujić P. Konveksni skupovi [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2017 [cited 2019 November 15] Available at: https://urn.nsk.hr/urn:nbn:hr:126:403773

P. Nujić, "Konveksni skupovi", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:126:403773

Metadata
TitleKonveksni skupovi
Title (english)Convex Sets
AuthorPetar Nujić
MentorMihaela Ribičić Penava (mentor)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Pure Mathematics)
(Algebra and Calculus Research Group)
Osijek
Defense date and country2017-12, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Mathematical Analysis
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.
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.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:403773
Study programmeTitle: Mathematics and Computer Science
Study programme type: university
Study level: integrated undergraduate and graduate
Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of usehttp://rightsstatements.org/vocab/InC/1.0/
Created on2017-12-08 09:08:27