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We apply Gromov's ham sandwich method to get  (1) domain monotonicity (up to a multiplicative constant factor);  (2) reverse domain monotonicity (up to a multiplicative constant factor); and  (3) universal inequalities  for Neumann eigenvalues of the Laplacian on bounded convex domains in aEuclidean space.

Applications of the `Ham Sandwich Theorem' to eigenvalues of the Laplacian