About: We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $/Omega$. We show that this size is strongly related to the regularity of $/partial /Omega$ by providing bounds on the Hausdorff dimension of $/partial E/cap /partial/Omega$. In particular we show that, if $/partial /Omega$ has $C^{1,/alpha}$ regularity then $/mathcal{H}^{d-2+/alpha}(/partial E/cap /partial/Omega)>0$. This shows that a sufficient condition to ensure that $/mathcal{H}^{d-1}(/partial E/cap /partial /Omega)>0$ is that $/partial /Omega$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $/partial E$ as well, we obtain that $/Omega$ convex, which yields $/partial E/in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.   Goto Sponge  NotDistinct  Permalink

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  • We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $/Omega$. We show that this size is strongly related to the regularity of $/partial /Omega$ by providing bounds on the Hausdorff dimension of $/partial E/cap /partial/Omega$. In particular we show that, if $/partial /Omega$ has $C^{1,/alpha}$ regularity then $/mathcal{H}^{d-2+/alpha}(/partial E/cap /partial/Omega)>0$. This shows that a sufficient condition to ensure that $/mathcal{H}^{d-1}(/partial E/cap /partial /Omega)>0$ is that $/partial /Omega$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $/partial E$ as well, we obtain that $/Omega$ convex, which yields $/partial E/in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.
Subject
  • Geometry
  • Concepts in logic
  • Concepts in metaphysics
  • Mathematical terminology
  • Necessity and sufficiency
  • Dichotomies
  • Conditionals
  • Fellows of the American Mathematical Society
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