Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results

Mabry, Rick; Deiermann, Paul

doi:10.4169/193009709x470317

Cited by 6 publications

(10 citation statements)

References 2 publications

(4 reference statements)

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“…Remark 7.7. If H is the product of the empty arrangement in R and a type I 2 (2k+1) arrangement in R 2 with k ≥ 2, then the corollary recovers the case of Entrée 2 on page 433 of [12] where N ≥ 5 is odd.…”

Section: And We Denote Bymentioning

confidence: 60%

“…This is known as the Pizza Theorem and was first stated as a problem in Mathematics Magazine by Upton [16] and solved by Goldberg [7]. There are many two-dimensional extensions of this result; see [6,12] and the references therein. An especially interesting solution to the original problem is by Carter and Wagon [4], who prove the result by a dissection.…”

Section: Introductionmentioning

confidence: 97%

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Sharing pizza in n dimensions

Ehrenborg,

Morel,

Readdy

2021

Preprint

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We introduce and prove the n-dimensional Pizza Theorem: Let H be a hyperplane arrangement in R n . If K is a measurable set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A n 1 such that the group of isometries W generated by the reflections in the hyperplanes of H contains the map − id, and if K is a translate of a convex body that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set K, the pizza quantity of K+a is polynomial in a for a small enough, for example if K is convex and 0 ∈ K + a. We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at a having radius R ≥ a vanishes for a Coxeter arrangement H with |H| − n an even positive integer. We also prove the Pizza Theorem for the surface volume: When H is a Coxeter arrangement and |H| − n is a nonnegative even integer, for an n-dimensional ball the alternating sum of the (n − 1)-dimensional surface volumes of the regions is equal to zero.

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Section: And We Denote Bymentioning

confidence: 60%

Section: Introductionmentioning

confidence: 97%

Sharing pizza in n dimensions

Ehrenborg,

Morel,

Readdy

2021

Preprint

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“…Remark 3.7. Theorem 3.4 immediately implies generalizations to our higher-dimensional case of the "thin crust" and "thick crust" results of Confection 3 and Leftovers 1 of [MD09] for an even number of cuts.…”

Section: A Dissection Proof Of the Higher-dimensional Pizza Theoremmentioning

confidence: 61%

“…Is it possible to give a dissection proof of this result? (2) Mabry and Deiermann [MD09] show that the two-dimensional pizza theorem does not hold for a dihedral arrangement having an odd number of lines. More precisely, they determine the sign of the quantity T (−1) T Vol(T ∩ K), where K is a disc containing the origin, and show that it vanishes if and only if the center of K lies on one of the lines.…”

Section: Let Us Mention Some Interesting Questions That Remain Openmentioning

confidence: 99%

Pizza and 2-structures

Ehrenborg,

Morel,

Readdy

2021

Preprint

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Let H be a Coxeter hyperplane arrangement on R n with Coxeter group W . We assume that the map − id is in the group W and that H is not of type A n 1 . Let K be a measurable subset of R n with finite volume such that K is stable by W and let a ∈ R n such that K contains the convex hull of {w(a) : w ∈ W }. The authors proved in a previous article that the alternating sum over the chambers T of H of the volumes of the sets T ∩ (K + a) is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity with the volume replaced by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from some basic geometry, the main ingredient is a previous theorem of the authors (the proof of which uses only properties of Coxeter systems and closed convex polyhedral cones) where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called 2-structures.

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“…For a pizza sliced as in FIGURE 1, Rick Mabry and Paul Deiermann showed that two people will get an equal amount of crust whenever they get an equal amount of pizza [10], whether the pizza has a "thin crust," represented by its boundary, or a "thick crust," represented by an annulus. Our results give an alternate, and no less elegant, proof of this fact for the thin-crust case: Our dissection of one set of slices gives pieces that we either move by translation, or flip over and then move, to give the other set of slices.…”

Section: Food For Further Thoughtmentioning

confidence: 99%

The Proof Is in the Pizza

Frederickson

2012

Mathematics Magazine

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Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results

Cited by 6 publications

References 2 publications

Sharing pizza in n dimensions

Sharing pizza in n dimensions

Pizza and 2-structures

The Proof Is in the Pizza

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