A note on the perimeter of fat objects
A number of realistic models have been studied earlier [see the survey article by \textit{M. de Berg, A. F. van der Stappen, J. Vleugels} and \textit{M. J. Katz}, Algorithmica 34, No.~1, 81--97 (2002; Zbl 1017.68141)]. Four such classes of objects are considered in the paper. The main result of the authors shows that \((\alpha, \beta)\)-covered objects have good perimeter length. Thus each point on the boundary of such an object sees a constant fraction of the length of its boundary. Further, \((\alpha, \beta)\)-covered objects are \(\varepsilon \)-boundary-good for some \(\varepsilon \) that depends only on \(\alpha\) and \(\beta\). It is also shown that a family of curves that converges to the Koch snowflake [see \textit{H. von Koch}, Arkiv f. Mat., Astr. och Fys. 1, 681--702 (1904; JFM 35.0387.02)], defines a family of objects that is locally \(\gamma\)-flat for a specific value of \(\gamma\), has diameter one, but contains objects of arbitrarily large perimeter length.
scientific article; zbMATH DE number 5800752