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:
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?
Is it always polar to $\alpha^+(t)$?
What is the relation between $\alpha^-(t)$ and $\Uparrow_{\alpha(t)}^A$
