Packing a Cake into a Box

@article{Skopenkov2011PackingAC,
  title={Packing a Cake into a Box},
  author={Mikhail Skopenkov},
  journal={The American Mathematical Monthly},
  year={2011},
  volume={118},
  pages={424 - 433}
}
  • M. Skopenkov
  • Published 10 March 2010
  • Physics
  • The American Mathematical Monthly
Abstract Given a triangular cake and a box in the shape of its mirror image, how can the cake be cut into a minimal number of pieces so that it can be put into the box? The cake has icing, so we are not allowed to put it into the box upside down. V. G. Boltyansky asked this question in 1977 and showed that three pieces always suffice. In this paper we provide examples of cakes that cannot be cut into two pieces to be put into the box. This shows that three is the answer to Boltyansky's question… 
Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing
TLDR
It is proved that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and a PTAS for minimizing the perimeter of the convex enclosure is given.

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