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

Abstract

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