By a unit cylinder <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\cal{C}}_{1}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">C</mi> </mrow> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> we mean the right-circular cylinder of radius one and height one. It is a convex solid (that is, a line segment joining any two points in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\cal{C}}_{1}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">C</mi> </mrow> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> lies entirely in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\cal{C}}_{1}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">C</mi> </mrow> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>), with volume π and surface area 4π. Thus, in relation to a unit sphere, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\cal{C}}_{1}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">C</mi> </mrow> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> has the same surface area but only three-fourths the volume.

The Biggest Convex Hull Within a Unit Cylinder

期刊论文
发表日期：

2025年9月1日

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Jyotirmoy Sarkar,
Collin Tully

摘要

By a unit cylinder ${\cal{C}}_{1}$ $C_{1}$ we mean the right-circular cylinder of radius one and height one. It is a convex solid (that is, a line segment joining any two points in ${\cal{C}}_{1}$ $C_{1}$ lies entirely in ${\cal{C}}_{1}$ $C_{1}$ ), with volume π and surface area 4π. Thus, in relation to a unit sphere, ${\cal{C}}_{1}$ $C_{1}$ has the same surface area but only three-fourths the volume.