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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?

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Can you cut and paste questions from assignments? ;-) If not, I apologise, but please have a read of mathoverflow.net/howtoask – David Roberts Oct 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 Petrunin Oct 22 '12 at 22:15

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