Magnituda metričnih prostorov : delo diplomskega seminarja

Rozman, Lara (author), Govc, Dejan (mentor), Lešnik, Davorin (comentor)

Magnituda je invarianta metričnih prostorov, ki jo definiramo za končne in kompaktne metrične prostore. Opazovani metrični prostor večkrat obravnavamo skupaj z družino njegovih raztegov, kar nam podaja magnitudno funkcijo. Limita magnitudne funkcije, ko prostor povsem skrčimo, je za nekatere razrede prostorov - npr. za prostore, ki jih lahko vložimo v $\mathbb{R}^n$ z 1-metriko ali evklidsko metriko - enaka 1. Temu pravimo, da ima prostor lastnost ene točke. Za prostore, ki nimajo lastnosti ene točke, velja, da je opazovana limita lahko poljubno realno število, ki je večje od 1.

• Language: Slovenian
• Keywords: magnituda, magnitudna funkcija, lastnost ene točke, utežitev, metrični prostor, pozitivno definiten prostor
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2026
• PID: 20.500.12556/RUL-188555
• UDC: 515.1
• COBISS.SI-ID: 292384515
• Publication date in RUL: 24.09.2026

Secondary language

• Language: English
• Title: Magnitude of metric spaces

Abstract

Magnitude is an invariant of metric spaces, defined for finite and compact metric spaces. The observed metric space is often considered alongside the family of its dilations, yielding the magnitude function. The limit of the magnitude function as the space is shrunk completely is equal to 1 for some classes of spaces - e.g. the ones embeddable into $\mathbb{R}^n$ with the 1-norm or the Euclidean norm. This is referred to as the space having the one-point property. For spaces lacking the one-point property, the limit in question can be any real number greater than 1.

• Keywords: magnitude, magnitude function, one-point property, weighting, metric space, positive definite space