The Monotonicity of the Cheeger constant for Parallel Bodies

Abstract We prove that for every planar convex set $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> , the function $$t\in (-r(\Omega ),+\infty )\longmapsto \sqrt{|\Omega _t|}h(\Omega _t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⟼</mml:mo> <mml:msqrt> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:msqrt> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is monotonically decreasing, where r , $$|\cdot |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>·</mml:mo> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> and h stand for the inradius, the measure and the Cheeger constant and $$(\Omega _t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for parallel bodies of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> . The result is shown not to hold when the convexity assumption is dropped. We also prove the differentiability of the map $$t\longmapsto h(\Omega _t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>⟼</mml:mo> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in any dimension and without any regularity assumption on the convex $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> , obtaining an explicit formula for the derivative. Those results are then combined to obtain estimates on the contact surface of the Cheeger sets of convex bodies. Finally, potential generalizations to other functionals such as the first eigenvalue of the Dirichlet Laplacian are explored.