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<p>The discrete isoperimetric inequality states that among all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( n \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-gons with a fixed area, the regular <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( n \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-gon has the least perimeter. We prove sharp bounds on the perimeter and area of hyperbolic tangential (respectively, cyclic) <i>n</i>-gons in terms of their inradius (respectively, circumradius), with extremality at the regular <i>n</i>-gon. We also establish sharp lower bounds on the total perimeter and total area of multiple tangential hyperbolic <i>n</i>-gons in terms of their total inradius.</p>

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Some analogues of isoperimetric inequality

  • 期刊论文

  • 发表日期：

    2025年12月13日

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  • Subash Chandra Behera,
  • Shiv Parsad

摘要

The discrete isoperimetric inequality states that among all \( n \) n -gons with a fixed area, the regular \( n \) n -gon has the least perimeter. We prove sharp bounds on the perimeter and area of hyperbolic tangential (respectively, cyclic) n-gons in terms of their inradius (respectively, circumradius), with extremality at the regular n-gon. We also establish sharp lower bounds on the total perimeter and total area of multiple tangential hyperbolic n-gons in terms of their total inradius.