distance from intersection of linear subspaces Consider two points  $x, y \in \mathbb{R}^n$, two linear subspaces $V_1, V_2 $ of $\mathbb{R}^n$ and a distance $d(\cdot , \cdot)$ in $\mathbb{R}^n$. Assume that $d(x,V_1) \leq d(y,V_1)$ and $d(x,V_2) \leq d(y,V_2)$. Is it true that $d(x,V_1 \cap V_2 ) \leq d(y,V_1\cap V_2)$ ? Note that the last inequality is in general false if $V_1$ and $V_2$ are arbitrary subsets.  In particular, I am interested in norm-induced distances.

it is not true. lets look at $\mathbb R^3$ , take $V_1$ be the $x,y$ plane and $V_2$ be the line connecting the points $(0,0,0)$ and $(2,0,1)$ and take $x=(2,0,\frac 1 2)$ and $y=(0,0,1)$. 
its esay to see that in the standard metric $d(x,V_1)=\frac 1 2$ and $d(x,V_2)\leq \frac1 2$  also $d(y,V_1)=1$ and $d(y,V_2)\geq \frac 1 2$ 
so indeed we get that $d(x,V_1)\leq d(y,V_1)$ and $d(x,V_2)\leq d(y,V_2)$
$V_1\cap V_2=\{(0,0,0)\}$  so we get $d(y,V_1\cap V_2)=1$ and $d(x,V_1\cap V_2)\geq2$ then $$d(y,V_1\cap V_2)\leq d(x,V_1\cap V_2)$$
