{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:35:07Z","timestamp":1776836107923,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"336","license":[{"start":{"date-parts":[[2023,3,30]],"date-time":"2023-03-30T00:00:00Z","timestamp":1680134400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100019217","name":"Institut de Valorisation des Donn\u00e9es","doi-asserted-by":"publisher","award":["PRF-2019-8079623546"],"award-info":[{"award-number":["PRF-2019-8079623546"]}],"id":[{"id":"10.13039\/501100019217","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 2 Superscript s\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=2^s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    vertices is not known when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s greater-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s \\ge 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3.24\">\n                        <mml:semantics>\n                          <mml:mn>3.24<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3.24%<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    larger than the width of the regular octagon:\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"cosine left-parenthesis pi slash 8 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>cos<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>8<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\cos (\\pi \/8)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In addition, the paper proposes a family of equilateral small\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -gons, for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 2 Superscript s\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=2^s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s greater-than-or-equal-to 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s\\ge 4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , whose widths are within\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis 1 slash n Superscript 4 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(1\/n^4)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the maximal width.\n                  <\/p>","DOI":"10.1090\/mcom\/3733","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T10:52:49Z","timestamp":1643799169000},"page":"2027-2040","source":"Crossref","is-referenced-by-count":1,"title":["The equilateral small octagon of maximal width"],"prefix":"10.1090","volume":"91","author":[{"given":"Christian","family":"Bingane","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Charles","family":"Audet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2022,3,30]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s00454-008-9103-9","article-title":"Isoperimetric polygons of maximum width","volume":"41","author":"Audet, Charles","year":"2009","journal-title":"Discrete Comput. 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A","ISSN":"https:\/\/id.crossref.org\/issn\/0097-3165","issn-type":"print"},{"issue":"4-5","key":"4","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1080\/10556780903087124","article-title":"Branching and bounds tightening techniques for non-convex MINLP","volume":"24","author":"Belotti, Pietro","year":"2009","journal-title":"Optim. Methods Softw.","ISSN":"https:\/\/id.crossref.org\/issn\/1055-6788","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/PL00000413","article-title":"On convex polygons of maximal width","volume":"74","author":"Bezdek, A.","year":"2000","journal-title":"Arch. Math. (Basel)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-889X","issn-type":"print"},{"key":"6","unstructured":"C. Bingane, Largest small polygons: a sequential convex optimization approach, Tech. Report G-2020-50, Les cahiers du GERAD,  arXiv:2009.07893, 2020."},{"key":"7","unstructured":"C. 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