Boundary Curvatures and the Distance Function

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)


Let \(\Omega \) be an open subset of \(\mathbb{R}^{n},\ n \geq 2,\) with non-empty boundary, and set
$$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}}$$
for the distance of \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega \). Our main objective in this chapter is to gather information about the regularity properties of δ.


Distance Function Boundary Curvature Principal Curvature Principal Direction Radon Measure 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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