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## What is $\alpha^-(t)$ for gradient curve in Alexandrov spaces?

Let $X$ be an $n$-dimensional Alexandrov space, $f: X\to \mathbb R$ a semi-concave function. One can define (see for example Petrunin "Semi-concave functions in Alexandrov geometry") the gradient curve of $f$ on a open set $U$ where $\nabla f\ne 0$ by: $$\alpha^+(t)=\nabla_{\alpha(t)}f$$ Since $\alpha$ is a lipschitz curve, according to Perelman-Petrunin "Quasi geodesic and gradient curves in Alexandrov spaces" section 2. In this case the left tangent vectors are unique for almost all $t$.

My question is:

1. Is there any simple example shows that $\alpha^-(t)$ fails to be unique for the semi-concave function of the type $d_A(\cdot)=dist(A, \cdot)$. Is there any general criterion gurantes the uniqueness for such left tangent vector?

2. Is it always polar to $\alpha^+(t)$?

3. What is the relation between $\alpha^-(t)$ and $\Uparrow_{\alpha(t)}^A$

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