Mikowski functional of the set $\{ (x,y) \in \mathbb R^2 \mid (x-1)^2 + y^2 \le 2, (x+1)^2 + y^2 \le 2 \}$

Question

I'm trying to find the explicit formula for the Mikowski functional of the set $$A = \{ (x,y) \in \mathbb R^2 \mid (x-1)^2 + y^2 \le 2, (x+1)^2 + y^2 \le 2 \}\,.$$ It's clear that $A = A_{-1} \cap A_{+1}$, where $$ A_{\pm 1} = \{ (x,y) \in \mathbb R^2 \mid (x \mp 1)^2 + y^2 \le 2 \}\, $$ and that Minkowski functional of an intersection can be expressed via Minkowski functionals of sets. I.e. $$ \mu(p, A) = \max \{ \mu(p, A_{-1}), \mu(p, A_{+1}) \}\,. $$ $A_{\pm 1}$ are disks and Minkowski functional of a disk with center in the origin is just the distance to the origin (up to multiplicative constant). But I don't see reasonable way to give the explicit formula for a Minkowski functional of translated disk.

Answer 1

$\mu(p,A_{+})=\inf \{t>0: (\frac x t, \frac y t) \in A_{+}\}$. This is $\inf \{t>0: (x+t)^{2}+y^{2} <2t^{2}\}=\inf \{t>0:t^{2}-2tx >x^{2}+y^{2}\}$. Hence $\mu(p,A_{+})=\inf \{t>0: (t-x)^{2} >2x^{2}+y^{2}\}=x+\sqrt {2x^{2}+y^{2}}$. $\mu(p,A_{-})$ is similar.