ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.09
https://doi.org/10.26493/1855-3974.2550.d96
(Also available at http://amc-journal.eu)
On generalized Minkowski arrangements*
Máté Kadlicskó
Department of Geometry, Budapest University of Technology,
Egry József utca 1., Budapest, Hungary, 1111
Zsolt Lángi †
MTA-BME Morphodynamics Research Group and Department of Geometry,
Budapest University of Technology,
Egry József utca 1., Budapest, Hungary, 1111
Received 9 Februar 2021, accepted 14 December 2021, published online 28 October 2022
Abstract
The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a
family of centrally symmetric convex bodies with the property that no member of the family
contains the center of any other member in its interior. This notion was generalized by Fejes
Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized
Minkowski arrangement of order µ for some 0 < µ < 1 if no member K of the family
overlaps the homothetic copy of any other member K ′ with ratio µ and with the same
center as K ′. In this note we prove a sharp upper bound on the total area of the elements
of a generalized Minkowski arrangement of order µ of finitely many circular disks in the
Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth
for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about
the maximum density of a generalized Minkowski arrangement of circular disks in the
plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski
arrangement of homothetic copies of a centrally symmetric convex body.
Keywords: Arrangement, Minkowski arrangement, density, homothetic copy.
Math. Subj. Class. (2020): 52C15, 52C26, 52A10
*The authors express their gratitude to K. Bezdek for directing their attention to this interesting problem, and
to two anonymous referees for many helpful suggestions. The second named author is supported by the National
Research, Development and Innovation Office, NKFI, K-119670, the János Bolyai Research Scholarship of the
Hungarian Academy of Sciences, and the BME IE-VIZ TKP2020 and ÚNKP-20-5 New National Excellence
Programs by the Ministry of Innovation and Technology.
†Corresponsing author.
E-mail addresses: kadlicsko.mate@gmail.com (Máté Kadlicskó), zlangi@math.bme.hu (Zsolt Lángi)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.09
https://doi.org/10.26493/1855-3974.2550.d96
(Dostopno tudi na http://amc-journal.eu)
Posplošeni sestavi Minkowskega*
Máté Kadlicskó
Department of Geometry, Budapest University of Technology,
Egry József utca 1., Budapest, Hungary, 1111
Zsolt Lángi †
MTA-BME Morphodynamics Research Group and Department of Geometry,
Budapest University of Technology,
Egry József utca 1., Budapest, Hungary, 1111
Prejeto 9. februarja 2021, sprejeto 14. decembra 2021, objavljeno na spletu 28. oktobra 2022
Povzetek
Pojem sestava Minkowskega je vpeljal Fejes Tóth leta 1965 kot družino središčno
simetričnih konveksnih teles z lastnostjo, da noben član te družine ne vsebuje središča
nobenega drugega člana v svoji notranjosti. Ta pojem je posplošil Fejes Tóth leta 1967, ko
je imenoval družino središčno simetričnih konveksnih teles posplošen sestav Minkowskega
reda µ za neki 0 < µ < 1, če noben član K te družine ne prekriva homotetične kopije
nobenega drugega člana K ′ z razmerjem µ in z istim središčem, kot ga ima K ′. V tem
članku dokažemo ostro zgornjo mejo za celotno površino elementov posplošenega sestava
Minkowskega reda µ, sestavljenega iz končno mnogo krožnih območij v evklidski ravnini.
Ta rezultat je posplošitev podobnega rezultata Fejesa Tótha za sestave Minkowskega iz
krožnih območij, kot tudi posplošitev rezultata Böröczkyja in Szabá o maksimalni gostoti
posplošenega sestava Minkowskega, sestoječega iz krožnih območij v ravnini. Izpeljemo
tudi ostro zgornjo mejo za gostoto posplošenega sestava Minkowskega, sestoječega iz ho-
motetičnih kopij središčno simetričnega konveksnega telesa.
Ključne besede: Sestav, sestav Minkowskega, gostota, homotetična kopija.
Math. Subj. Class. (2020): 52C15, 52C26, 52A10
*Avtorja izražata svojo hvaležnost K. Bezdeku za usmeritev njune pozornosti na ta zanimivi problem, in dvema
neznanima recenzentoma za mnoge koristne predloge. Drugi avtor je podprt s strani National Research, Develop-
ment and Innovation Office, NKFI, K-119670, the János Bolyai Research Scholarship of the Hungarian Academy
of Sciences, s strani BME IE-VIZ TKP2020 ter s strani ÚNKP-20-5 New National Excellence Programs by the
Ministry of Innovation and Technology.
†Kontaktni avtor.
E-poštna naslova: kadlicsko.mate@gmail.com (Máté Kadlicskó), zlangi@math.bme.hu (Zsolt Lángi)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/