Counterexamples to the Knaster problem in all remaining dimensions

Abstract

We construct counterexamples to Knaster's 1947 problem in all remaining cases. For every ambient dimension n ≥ 4, we obtain a real polynomial on R^{n} and n linearly independent points on S^{n-1} whose values cannot be equalized by any orthogonal transformation. These examples settle the remaining cases with target dimension m = 1 and, together with earlier results, complete the determination of the parameters in the original problem. The construction begins with a stronger four-point theorem on every sphere S^{n-1}, n ≥ 2. Its main ingredient is a suspension theorem that extends Karasev's local obstruction to arbitrary dimension by preserving a differential condition for an even function and an odd function. A uniform Taylor estimate yields finite configurations, and stability under perturbation gives the polynomial examples. The four-point theorem also disproves the uniform weak Knaster conjecture for real-valued functions.