We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem in the proof of our second theorem that if C is a two-dimensional spherical convex body of diameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2494_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta \in (\frac{1}{2}\pi ,\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then the diameter of the set of extreme points of C is at least <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2494_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="226" /> </InlineMediaObject> <EquationSource Format="TEX">$2 \arccos \big (\frac{1}{4}(\cos \delta + \sqrt{\cos ^2 \delta +8})\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>arccos</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mo stretchy="false">(</mo> <mo>cos</mo> <mi>δ</mi> <mo>+</mo> <msqrt> <mrow> <msup> <mo>cos</mo> <mn>2</mn> </msup> <mi>δ</mi> <mo>+</mo> <mn>8</mn> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This estimate cannot be improved.

Spherical Quadrilateral with Three Right Angles and Its Application for Diameter of Extreme Points of a Convex Body

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2025年8月13日

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Marek Lassak

摘要

We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem in the proof of our second theorem that if C is a two-dimensional spherical convex body of diameter $\delta \in (\frac{1}{2}\pi ,\pi )$ $δ \in (\frac{1}{2} π, π)$ , then the diameter of the set of extreme points of C is at least $2 \arccos \big (\frac{1}{4}(\cos \delta + \sqrt{\cos ^2 \delta +8})\big )$ $2 arccos (\frac{1}{4} (cos δ + \sqrt{{cos}^{2} δ + 8}))$ . This estimate cannot be improved.