Let $M$ be an n-dim Alexandrov space with curvature bounded below $sec \geqslant k$, possibly non-compact. We assume that $M$ has no boundary for simplicity. For a compact subset $K \subset M$, the cut points $C_K$ is defined as the points $x$ such that any geodesics connecting $K$ and $x$ are not extendable.

Let $d_K$ be the distance function to $K$. We know that $d_K$ is semiconcave on $$(M\K)\cap\{d_K < \frac{\pi}{2\sqrt k}\}$$.

For a semiconcave function $f$, the gradient $\nabla_x f=d_xf(\xi_{max})\cdot \xi_{max}$ exists, possible 0. Then at $$M\(K\cap C_K)\cap\{d_K < \frac{\pi}{2\sqrt k}\}, |\nabla_x d_K|=1.$$ But at cut point $x$, $\nabla_x d_K$ may be 0 (for example, the local maximal point for $d_K$). Otherwise not 0, but $|\nabla_x d_K|<1?$ So can one give an example for the last case?

Begging at "good" point, the gradient flow is a unit speed geodesic for a short time, and stop or change direction when meeting cut point.

Let $M$ be an $n$-dim Alexandrov space with curvature bounded below $sec \geqslant k$, possibly non-compact. We assume that $M$ has no boundary for simplicity. For a compact subset $K \subset M$, the cut points $C_K$ are defined as the points $x$ such that any geodesics connecting $K$ and $x$ are not extendable.

Let $d_K$ be the distance function to $K$. We know that $d_K$ is semiconcave on $$(M \setminus K)\cap\{d_K < \frac{\pi}{2\sqrt k}\}.$$

For a semiconcave function $f$, the gradient $\nabla_x f=d_xf(\xi_{max})\cdot \xi_{max}$ exists, possibly $0$. Then at $$M \setminus (K\cap C_K)\cap\{d_K < \frac{\pi}{2\sqrt k}\}, |\nabla_x d_K|=1.$$ But at cut point $x$, $\nabla_x d_K$ may be $0$ (for example, the local maximal point for $d_K$). Otherwise not $0$, but $|\nabla_x d_K|<1?$ So can one give an example for this last case?

Beginning at a "good" point, the gradient flow is a unit speed geodesic for a short time, and stops or changes direction when meeting a cut point.