Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a totally convex set $C\in M^n$, we mean for any two point $a, b\in C$ and any geodesic joining them must lie in $A$. In order to define a retraction from $^aC$ to $C$, where $^aC$ is defined by $$ ^a C=x\in M, d(x, C) < a.$$ Here $a$ must be small enough such that for any $x\in ^aC\setminus C$ there is a unique point $h(a)\in C$ such that $d(x, C)=d(x, h(x))$.

I am wondering whether there is a example showing that when $a$ is large, the projection $h$ is not well defined, i.e. there are two points $p, q\in C$ such that $d(x, p)=d(x, q)=d(x, C)$.

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