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A Sharp Quantitative Estimate for the Perimeters of Convex Sets in the Plane

M. Carozza F. Giannetti F. Leonetti Antonia Passarelli di Napoli

2015 · Journal of Convex Analysis · 3 citations · ⚡ 2 influential · 3 references

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">E \subset B \subset \mathbb R^2</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>B</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be bounded, convex sets. The monotonicity of the perimeters holds, i.e. <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathcal{H}^{1} (\partial E) \leqslant \mathcal{H}^{1}(\partial B)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="script">H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⩽</mml:mo> <mml:msup> <mml:mi mathvariant="script">H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . Here we give a quantitative estimate of the difference of the perimeters: it shows how much the perimeter of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">B</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> increases when <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">B</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> becomes larger and larger with respect to <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">E</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We give an example showing that our estimate is sharp. </jats:p>

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