On nearly optimal paper Moebius bands

Abstract Let ε &lt; 1/324 and let Ω be a smooth embedded paper Moebius band of aspect ratio less than <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msqrt> <m:mn>3</m:mn> </m:msqrt> <m:mo>+</m:mo> <m:mi>ε</m:mi> <m:mo>.</m:mo> </m:math> $\sqrt{3}+\varepsilon .$ We prove that Ω is within Hausdorff distance <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>18</m:mn> <m:msqrt> <m:mi>ε</m:mi> </m:msqrt> </m:math> $18 \sqrt{\varepsilon}$ of an equilateral triangle of perimeter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>2</m:mn> <m:msqrt> <m:mn>3</m:mn> </m:msqrt> <m:mo>.</m:mo> </m:math> $2 \sqrt{3} .$ This is an effective and fairly sharp version of our recent theorems in [6] about the optimal paper Moebius band.