A short note on the Schiffer's conjecture for a class of centrally symmetric convex domains in ${\mathbb{R}}^2$

Let $\mathrm{\Omega }$ be a bounded centrally symmetric domain in ${\mathbb{R}}^2$ with analytic boundary $\mathrm{\partial }\mathrm{\Omega }$ and center $c$. Let $\tau =\tau (\mathrm{\Omega })$ be the number of points $p$ on $\mathrm{\partial }\mathrm{\Omega }$ such that the normal line to $\mathrm{\partial }\mathrm{\Omega }$ at $p$ passes through $c$. We show that if $\tau <8$ then $\mathrm{\Omega }$ satisfies the Schiffer’s conjecture.