I have the following question:

Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$ (via infium of lenght of curves in X connecting two points). Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact? The problem is of course that curves connecting points in $A$ may not entirely lie in $A$.

Best wishes, Mike

Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$ (via infimum of length of curves in X connecting two points). 

Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact? The problem is of course that curves connecting points in $A$ may not entirely lie in $A$.

I have the following question:

Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$. Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact?

Best wishes, Mike

I have the following question:

Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$ (via infium of lenght of curves in X connecting two points). Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact? The problem is of course that curves connecting points in $A$ may not entirely lie in $A$.

Best wishes, Mike

I have the following question:

Assume $g$ and $h$ are smooth Riemannian metrics on manifold X with $d_g$ and $d_h$ the induced distance functions on $X$. Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact?

Best wishes, Mike

I have the following question:

Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$. Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact?

Best wishes, Mike