Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteen

In dimensions $3,4,6,8,10,14$ we construct bounded convex non-ball domains with real-analytic spherical boundaries. Each admits a nonconstant solution of $Δu+u=0$ with $u=1$ and $\nabla u=0$ on the boundary. Green's identity makes their indicator Fourier transforms vanish on the unit sphere, disprov