Positivity principle for measures on uniformly convex Banach spaces

A Banach space X X is said to satisfy the positivity principle for small balls if for every finite Borel measures μ \mu and ν \nu on X X , the inequalities μ ( B ) ≤ ν ( B ) \mu (B) \leq \nu (B) for all balls B of radius less than 1 imply that μ ≤ ν \mu \leq \nu . It is shown that no uniformly convex infinite-dimensional separable Banach space X X obeys the positivity principle for small balls.