Exercise 14, § 3

Let $E$ be a closed subset of $\mathbf{R}^n$. Suppose that there exists an interior point $p$ of E such that every line passing through $p$ meets the boundary of E in at most two points. Then there exists only one closed ideal I of $L^1(\mathbf{R}^n)$ such that $\mathscr{F}(f)$ vanishes on E. (Consider $f$ as a limit in $L^1(\mathbf{R}^n)$ of functions deduced from $f$ by homotheties with center $p$).