Counting blanks in polygonal arrangements

Arseniy Akopyan, Erel Segal-Halevi

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.

Original languageEnglish
Pages (from-to)2242-2257
Number of pages16
JournalSIAM Journal on Discrete Mathematics
Issue number3
StatePublished - 2018


  • Cake-cutting
  • Convex objects
  • Covering
  • Packing
  • Rectilinear polygons
  • Redivision


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