Mass of rays on Alexandrov surfaces (Q2769877)

Let \(X\) be a complete, connected and finitely connected Alexandrov space without boundary of dimension two whose curvature is bounded below by a constant and admitting total curvature. Denote by \(F_p\) the union of all rays in \(X\) emanating from the point \(p\in X\) and by \(A(p)\) the set of the directions of rays at \(p\). One of the two results of this paper is the following: Every \(p\in X\) has the property that \(\mu(A(p))= 2\pi\chi (X)-C(X-F_p)\), where \(\mu(A(p))\) is the Lebesgue measure of \(A(p)\), \(\chi(X)\) is the Euler characteristic of \(X\) and \(C(U)\) denotes the total excess of the subset \(U\subset X\). The Riemannian case of this was proved by \textit{K. Shiga} [Tsukuba J. Math. 6, 41-50 (1982; Zbl 0531.53035)].