A note on starshaped sets in 2-dimensional manifolds without conjugate points (Q403256)

Summary: Let \(W^n\) be \(C^\infty\)-complete, simply connected \(n\)-dimensional Riemannian manifolds without conjugate points. Assume that \(n=2\) and \(S\subset W^2\) is starshaped where \(\mathrm{ker}S\neq S\). For every point \(x\in S\backslash\mathrm{ker}S\), define \(A(x)=\{y:y\) lies on some geodesic segment in \(S\) from \(x\) to a point of \(\mathrm{ker}S\}\). There is a finite collection \(\mathcal{A}\) of all maximal \(A\) sets whose union is \(S\). Further, \(\mathrm{ker} S=\cap\{ A:A\text{ in }\mathcal{A}\}\).