Generalised Flatness Constants: A Framework Applied in Dimension $2$. Codenotti, G., Hall, T., & Hofscheier, J. September, 2021. ![Generalised Flatness Constants: A Framework Applied in Dimension $2$ [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper abstract bibtex Let $A {\}in {\}\{ {\}mathbb\{Z\}, {\}mathbb\{R\} {\}\}$ and $X {\}subset {\}mathbb\{R\}{\textasciicircum}d$ be a bounded set. Affine transformations given by an automorphism of ${\}mathbb\{Z\}{\textasciicircum}d$ and a translation in $A{\textasciicircum}d$ are called (affine) $A$-unimodular transformations. The image of $X$ under such a transformation is called an $A$-unimodular copy of $X$. It was shown in [Averkov, Hofscheier, Nill, 2019] that every convex body whose width is "big enough" contains an $A$-unimodular copy of $X$. The threshold when this happens is called the generalised flatness constant ${\}mathrm\{Flt\}_d{\textasciicircum}A(X)$. It resembles the classical flatness constant if $A={\}mathbb\{Z\}$ and $X$ is a lattice point. In this work, we introduce a general framework for the explicit computation of these numerical constants. The approach relies on the study of $A$-$X$-free convex bodies generalising lattice-free (also known as hollow) convex bodies. We then focus on the case that $X=P$ is a full-dimensional polytope and show that inclusion-maximal $A$-$P$-free convex bodies are polytopes. The study of those inclusion-maximal polytopes provides us with the means to explicitly determine generalised flatness constants. We apply our approach to the case $X={\}Delta_2$ the standard simplex in ${\}mathbb\{R\}{\textasciicircum}2$ of normalised volume $1$ and compute ${\}mathrm\{Flt\}{\textasciicircum}\{{\}mathbb\{R\}\}_2({\}Delta_2)=2$ and ${\}mathrm\{Flt\}{\textasciicircum}\{{\}mathbb\{Z\}\}_2({\}Delta_2)={\}frac\{10\}3$.
@article{codenotti_generalised_2021,
title = {Generalised {Flatness} {Constants}: {A} {Framework} {Applied} in {Dimension} \$2\$},
shorttitle = {Generalised {Flatness} {Constants}},
url = {https://arxiv.org/abs/2110.02770v1},
abstract = {Let \$A {\textbackslash}in {\textbackslash}\{ {\textbackslash}mathbb\{Z\}, {\textbackslash}mathbb\{R\} {\textbackslash}\}\$ and \$X {\textbackslash}subset {\textbackslash}mathbb\{R\}{\textasciicircum}d\$ be a bounded set. Affine transformations given by an automorphism of \${\textbackslash}mathbb\{Z\}{\textasciicircum}d\$ and a translation in \$A{\textasciicircum}d\$ are called (affine) \$A\$-unimodular transformations. The image of \$X\$ under such a transformation is called an \$A\$-unimodular copy of \$X\$. It was shown in [Averkov, Hofscheier, Nill, 2019] that every convex body whose width is "big enough" contains an \$A\$-unimodular copy of \$X\$. The threshold when this happens is called the generalised flatness constant \${\textbackslash}mathrm\{Flt\}\_d{\textasciicircum}A(X)\$. It resembles the classical flatness constant if \$A={\textbackslash}mathbb\{Z\}\$ and \$X\$ is a lattice point. In this work, we introduce a general framework for the explicit computation of these numerical constants. The approach relies on the study of \$A\$-\$X\$-free convex bodies generalising lattice-free (also known as hollow) convex bodies. We then focus on the case that \$X=P\$ is a full-dimensional polytope and show that inclusion-maximal \$A\$-\$P\$-free convex bodies are polytopes. The study of those inclusion-maximal polytopes provides us with the means to explicitly determine generalised flatness constants. We apply our approach to the case \$X={\textbackslash}Delta\_2\$ the standard simplex in \${\textbackslash}mathbb\{R\}{\textasciicircum}2\$ of normalised volume \$1\$ and compute \${\textbackslash}mathrm\{Flt\}{\textasciicircum}\{{\textbackslash}mathbb\{R\}\}\_2({\textbackslash}Delta\_2)=2\$ and \${\textbackslash}mathrm\{Flt\}{\textasciicircum}\{{\textbackslash}mathbb\{Z\}\}\_2({\textbackslash}Delta\_2)={\textbackslash}frac\{10\}3\$.},
language = {en},
urldate = {2021-10-12},
author = {Codenotti, Giulia and Hall, Thomas and Hofscheier, Johannes},
month = sep,
year = {2021},
keywords = {differential geometry, mentions sympy, symplectic manifolds},
}
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The threshold when this happens is called the generalised flatness constant ${\\}mathrm\\{Flt\\}_d{\\textasciicircum}A(X)$. It resembles the classical flatness constant if $A={\\}mathbb\\{Z\\}$ and $X$ is a lattice point. In this work, we introduce a general framework for the explicit computation of these numerical constants. The approach relies on the study of $A$-$X$-free convex bodies generalising lattice-free (also known as hollow) convex bodies. We then focus on the case that $X=P$ is a full-dimensional polytope and show that inclusion-maximal $A$-$P$-free convex bodies are polytopes. The study of those inclusion-maximal polytopes provides us with the means to explicitly determine generalised flatness constants. We apply our approach to the case $X={\\}Delta_2$ the standard simplex in ${\\}mathbb\\{R\\}{\\textasciicircum}2$ of normalised volume $1$ and compute ${\\}mathrm\\{Flt\\}{\\textasciicircum}\\{{\\}mathbb\\{R\\}\\}_2({\\}Delta_2)=2$ and ${\\}mathrm\\{Flt\\}{\\textasciicircum}\\{{\\}mathbb\\{Z\\}\\}_2({\\}Delta_2)={\\}frac\\{10\\}3$.","language":"en","urldate":"2021-10-12","author":[{"propositions":[],"lastnames":["Codenotti"],"firstnames":["Giulia"],"suffixes":[]},{"propositions":[],"lastnames":["Hall"],"firstnames":["Thomas"],"suffixes":[]},{"propositions":[],"lastnames":["Hofscheier"],"firstnames":["Johannes"],"suffixes":[]}],"month":"September","year":"2021","keywords":"differential geometry, mentions sympy, symplectic manifolds","bibtex":"@article{codenotti_generalised_2021,\n\ttitle = {Generalised {Flatness} {Constants}: {A} {Framework} {Applied} in {Dimension} \\$2\\$},\n\tshorttitle = {Generalised {Flatness} {Constants}},\n\turl = {https://arxiv.org/abs/2110.02770v1},\n\tabstract = {Let \\$A {\\textbackslash}in {\\textbackslash}\\{ {\\textbackslash}mathbb\\{Z\\}, {\\textbackslash}mathbb\\{R\\} {\\textbackslash}\\}\\$ and \\$X {\\textbackslash}subset {\\textbackslash}mathbb\\{R\\}{\\textasciicircum}d\\$ be a bounded set. Affine transformations given by an automorphism of \\${\\textbackslash}mathbb\\{Z\\}{\\textasciicircum}d\\$ and a translation in \\$A{\\textasciicircum}d\\$ are called (affine) \\$A\\$-unimodular transformations. The image of \\$X\\$ under such a transformation is called an \\$A\\$-unimodular copy of \\$X\\$. It was shown in [Averkov, Hofscheier, Nill, 2019] that every convex body whose width is \"big enough\" contains an \\$A\\$-unimodular copy of \\$X\\$. The threshold when this happens is called the generalised flatness constant \\${\\textbackslash}mathrm\\{Flt\\}\\_d{\\textasciicircum}A(X)\\$. It resembles the classical flatness constant if \\$A={\\textbackslash}mathbb\\{Z\\}\\$ and \\$X\\$ is a lattice point. In this work, we introduce a general framework for the explicit computation of these numerical constants. The approach relies on the study of \\$A\\$-\\$X\\$-free convex bodies generalising lattice-free (also known as hollow) convex bodies. We then focus on the case that \\$X=P\\$ is a full-dimensional polytope and show that inclusion-maximal \\$A\\$-\\$P\\$-free convex bodies are polytopes. The study of those inclusion-maximal polytopes provides us with the means to explicitly determine generalised flatness constants. We apply our approach to the case \\$X={\\textbackslash}Delta\\_2\\$ the standard simplex in \\${\\textbackslash}mathbb\\{R\\}{\\textasciicircum}2\\$ of normalised volume \\$1\\$ and compute \\${\\textbackslash}mathrm\\{Flt\\}{\\textasciicircum}\\{{\\textbackslash}mathbb\\{R\\}\\}\\_2({\\textbackslash}Delta\\_2)=2\\$ and \\${\\textbackslash}mathrm\\{Flt\\}{\\textasciicircum}\\{{\\textbackslash}mathbb\\{Z\\}\\}\\_2({\\textbackslash}Delta\\_2)={\\textbackslash}frac\\{10\\}3\\$.},\n\tlanguage = {en},\n\turldate = {2021-10-12},\n\tauthor = {Codenotti, Giulia and Hall, Thomas and Hofscheier, Johannes},\n\tmonth = sep,\n\tyear = {2021},\n\tkeywords = {differential geometry, mentions sympy, symplectic manifolds},\n}\n\n\n\n\n\n\n\n","author_short":["Codenotti, G.","Hall, T.","Hofscheier, J."],"key":"codenotti_generalised_2021","id":"codenotti_generalised_2021","bibbaseid":"codenotti-hall-hofscheier-generalisedflatnessconstantsaframeworkappliedindimension2-2021","role":"author","urls":{"Paper":"https://arxiv.org/abs/2110.02770v1"},"keyword":["differential geometry","mentions sympy","symplectic manifolds"],"metadata":{"authorlinks":{}}},"bibtype":"article","biburl":"https://bibbase.org/zotero-group/nicoguaro/525293","dataSources":["YtBDXPDiQEyhyEDZC","fhHfrQgj3AaGp7e9E","qzbMjEJf5d9Lk78vE","45tA9RFoXA9XeH4MM","MeSgs2KDKZo3bEbxH","nSXCrcahhCNfzvXEY","ecatNAsyr4f2iQyGq","tpWeaaCgFjPTYCjg3"],"keywords":["differential geometry","mentions sympy","symplectic manifolds"],"search_terms":["generalised","flatness","constants","framework","applied","dimension","codenotti","hall","hofscheier"],"title":"Generalised Flatness Constants: A Framework Applied in Dimension $2$","year":2021}