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In paragraph four, the gradient should be measured at $q$, not $x$. Paragraph 5 is edited to clarify where $x$ is allowed vary.
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John Harvey
John Harvey
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The distance function is semi-concave on all of $M \setminus K$. The additional restriction is unnecessary.

The easiest example I can think of for a gradient strictly between zero and one is the cone. Consider a 2-dimensional cone, and let $K = \lbrace p \rbrace$ be a single point which is not the vertex.

Let $q$ be the point antipodal to $p$, by which I mean the point at the same distance from the origin as $p$, but as far away from $p$ as possible.

Now there are two geodesics from $p$ to $q$, so extension is not possible. The vector $\xi_{\textrm{max}}$ is the vector at $q$ which points away from the origin, and $0 < \left| \nabla_x d_K \right| < 1$$0 < \left| \nabla_q d_K \right| < 1$.

In fact, varying $x$ along the ray from the origin through $q$, one can obtain every possible value $0 \leq \left| \nabla_x d_K \right| \leq 1$. You can see this by cutting the cone along the ray through $p$, and considering the two geodesics from $p$ to any point of the ray.

The distance function is semi-concave on all of $M \setminus K$. The additional restriction is unnecessary.

The easiest example I can think of for a gradient strictly between zero and one is the cone. Consider a 2-dimensional cone, and let $K = \lbrace p \rbrace$ be a single point which is not the vertex.

Let $q$ be the point antipodal to $p$, by which I mean the point at the same distance from the origin as $p$, but as far away from $p$ as possible.

Now there are two geodesics from $p$ to $q$, so extension is not possible. The vector $\xi_{\textrm{max}}$ is the vector at $q$ which points away from the origin, and $0 < \left| \nabla_x d_K \right| < 1$.

In fact, along the ray from the origin through $q$, one can obtain every possible value $0 \leq \left| \nabla_x d_K \right| \leq 1$. You can see this by cutting the cone along the ray through $p$, and considering the two geodesics from $p$ to any point of the ray.

The distance function is semi-concave on all of $M \setminus K$. The additional restriction is unnecessary.

The easiest example I can think of for a gradient strictly between zero and one is the cone. Consider a 2-dimensional cone, and let $K = \lbrace p \rbrace$ be a single point which is not the vertex.

Let $q$ be the point antipodal to $p$, by which I mean the point at the same distance from the origin as $p$, but as far away from $p$ as possible.

Now there are two geodesics from $p$ to $q$, so extension is not possible. The vector $\xi_{\textrm{max}}$ is the vector at $q$ which points away from the origin, and $0 < \left| \nabla_q d_K \right| < 1$.

In fact, varying $x$ along the ray from the origin through $q$, one can obtain every possible value $0 \leq \left| \nabla_x d_K \right| \leq 1$. You can see this by cutting the cone along the ray through $p$, and considering the two geodesics from $p$ to any point of the ray.

Source Link
John Harvey
John Harvey
  • 599
  • 2
  • 11

The distance function is semi-concave on all of $M \setminus K$. The additional restriction is unnecessary.

The easiest example I can think of for a gradient strictly between zero and one is the cone. Consider a 2-dimensional cone, and let $K = \lbrace p \rbrace$ be a single point which is not the vertex.

Let $q$ be the point antipodal to $p$, by which I mean the point at the same distance from the origin as $p$, but as far away from $p$ as possible.

Now there are two geodesics from $p$ to $q$, so extension is not possible. The vector $\xi_{\textrm{max}}$ is the vector at $q$ which points away from the origin, and $0 < \left| \nabla_x d_K \right| < 1$.

In fact, along the ray from the origin through $q$, one can obtain every possible value $0 \leq \left| \nabla_x d_K \right| \leq 1$. You can see this by cutting the cone along the ray through $p$, and considering the two geodesics from $p$ to any point of the ray.