The equilateral small octagon of maximal width

A small polygon is a of unit diameter. The maximal width an equilateral with n = 2 s n=2^s vertices not known when alttext="s greater-than-or-equal-to 3"> ≥ 3 encoding="application/x-tex">s \ge 3 . This paper solves the first open case and finds optimal octagon. Its approximately alttext="3.24"> 3.24 encoding="application/x-tex">3.24% larger than regular octagon: alttext="cosine left-parenthesis pi slash 8 right-parenthesis"> cos ⁡ / 8 stretchy="false">) encoding="application/x-tex">\cos (\pi /8) In addition, proposes family alttext="n"> encoding="application/x-tex">n -gons, for 4"> 4 encoding="application/x-tex">s\ge 4 , whose widths are within alttext="upper O 1 n 4 Baseline O 1 encoding="application/x-tex">O(1/n^4) width.