Besicovitch-type properties of measures and submanifolds. (English) Zbl 0618.53006

Kowalski, Oldřich; Preiss, David

J. Reine Angew. Math. 379, 115-151 (1987).

One aim of this paper is to classify all \(C^1\)-hypersurfaces \(M\subset R^{n+1}\) with the following property: for each \(x\in M\) and each radius \(r>0\), the intersection of the \((n+1)\)-dimensional ball \(B(x;r)\) with \(M\) has the same \(n\)-dimensional (Riemannian) volume as the intersection of \(B(x;r)\) with the tangent hyperplane \(T_xM\). We prove that each such hypersurface is congruent to an open dense subset in \(R^n\subset R^{n+1}\), or in \(C^3_1\times R^{n-3}\), where \(C^3_1\) is the “light cone” \((x_4)^2=(x_1)^2+(x_2)^2+(x_3)^2\) of \(R^4\).
More generally, we classify all non-zero Borel measures \(\Phi\) on \(R^{n+1}\) such that \(\Phi (B(x;r)) = \alpha_nr^n\) holds for each \(x\in \text{spt }\Phi\) and each \(r>0\) (where \(\alpha_ n\) denotes the volume of the unit ball in \(R^n)\). We call the last property “global Besicovitch property” because the second author has used Borel measures of this type in his recent solution of an outstanding conjecture stimulated by the old work of A. S. Besicovitch [cf. H. Federer, Geometric measure theory. Berlin etc.: Springer-Verlag (1969; Zbl 0176.00801), paragraph 3.3.22].

Reviewer: Oldřich Kowalski

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
28A75	Length, area, volume, other geometric measure theory
49Q15	Geometric measure and integration theory, integral and normal currents in optimization

Keywords:

volume; hypersurface; Borel measures; Besicovitch property

Citations:

Zbl 0176.00801