M is an Alexandrov space with curv>=-1,containing a line(ray).Is level set of Busemann function on M again Alexandrov space?If not,can you give a counterexample?

Can you cut and paste questions from assignments? ;-) If not, I apologise, but please have a read of mathoverflow.net/howtoask
– David RobertsOct 22 '12 at 8:31

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No, consider the quotient $\mathbb H^3/\iota$, where $\iota$ is rotation around a line.
– Anton PetruninOct 22 '12 at 22:15