Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions two to eighteen

Paper and computer-assisted proofs (Arb ball arithmetic plus explicit analytic tail bounds) of bounded convex non-ball domains in R^n, for every n from 2 to 18 and for n = 20, 21, with real-analytic boundaries diffeomorphic to spheres, carrying nonconstant solutions of Delta u + u = 0 with u = 1 and grad u = 0 on the boundary. These are counterexamples to the Schiffer conjecture and to the Pompeiu conjecture within the class of convex domains; the planar domain is the first convex planar counterexample. The paper also contains a strictly convex planar domain for Berenstein's problem (u = 0 and constant normal derivative), and a second, strictly star-shaped non-convex example in R^3. The constructions, proofs, verification code and text were produced with AI models (Claude Opus 5.5, Claude Sonnet 5.5, GPT-6 Astra, GPT-6 Sol and GPT-6.1 Sol) under the author's direction.