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- W2950273625 abstract "We construct the first known hollow lattice polytopes of width larger than dimension: a hollow lattice polytope (resp., a hollow lattice simplex) of dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=14> <mml:semantics> <mml:mn>14</mml:mn> <mml:annotation encoding=application/x-tex>14</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp., <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=404> <mml:semantics> <mml:mn>404</mml:mn> <mml:annotation encoding=application/x-tex>404</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) and of width <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=15> <mml:semantics> <mml:mn>15</mml:mn> <mml:annotation encoding=application/x-tex>15</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp., <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=408> <mml:semantics> <mml:mn>408</mml:mn> <mml:annotation encoding=application/x-tex>408</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). We also construct a hollow (nonlattice) tetrahedron of width <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 plus StartRoot 2 EndRoot> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding=application/x-tex>2+sqrt 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we conjecture that this is the maximum width among <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, the constructions above imply the existence of hollow lattice polytopes (resp., hollow simplices) of arbitrarily large dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and width <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=asymptotically-equals 1.14 d> <mml:semantics> <mml:mrow> <mml:mo>≃<!-- ≃ --></mml:mo> <mml:mn>1.14</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>simeq 1.14 d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp., <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=asymptotically-equals 1.01 d> <mml:semantics> <mml:mrow> <mml:mo>≃<!-- ≃ --></mml:mo> <mml:mn>1.01</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>simeq 1.01 d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>)." @default.
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- W2950273625 date "2019-08-07" @default.
- W2950273625 modified "2023-10-17" @default.
- W2950273625 title "Hollow polytopes of large width" @default.
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