Gromov–Hausdorff distances from simply connected geodesic spaces to the circle

We prove that the Gromov–Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction pi Over 4 EndFraction"> <mml:semantics> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mn>4</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {\pi }{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also prove that this bound is tight through the construction of a simply connected geodesic space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper E"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which attains the lower bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction pi Over 4 EndFraction"> <mml:semantics> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mn>4</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {\pi }{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We deduce the first statement from a general result that we also establish which gives conditions on how small the Gromov–Hausdorff distance between two geodesic metric spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma d Subscript upper X Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X, d_X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper Y comma d Subscript upper Y Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(Y, d_Y )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has to be in order for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _1(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 left-parenthesis upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _1(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be isomorphic.