On the Weyl tube formula for nonlocal perimeters

Abstract

We study the asymptotic behaviour of the nonlocal perimeter of tubular neighbourhoods. For a closed $C^{1,1}$ submanifold of Euclidean space, we show that a suitable rescaling of the nonlocal perimeter of its tube converges to an explicit constant times the Hausdorff measure of the submanifold. The constant is the fractional perimeter of the unit ball in the codimension of the submanifold. For finite point sets, we obtain an exact tube formula in terms of a generalised hypergeometric function. In contrast with the classical Weyl formula, this expansion has infinitely many terms, contains both noninteger and integer powers of the tubular radius, and has coefficients that depend on the relative positions of the points rather than only on their intrinsic geometry. We also establish a point-set expansion for general analytic kernels.