Gromov's systolic inequality asserts that the length, sys_1(M~n), of the shortest noncontractible curve in a closed essential Riemannian manifold M~n does not exceed c(n) vol~(1/n)(M~n) for some constant c(n). (Essential manifolds is a class of non-simply connected manifolds that includes all non-simply connected closed surfaces, tori and projective spaces.) Here we prove that all closed essential Riemannian manifolds satisfy sys_1(M~n) ≤ n vol~(1/n) (M~n). (The best previously known upper bound for c(n) was exponential in n.) We similarly improve a number of related inequalities. We also give a qualitative strengthening of Guth's theorem (2011, 2017) asserting that if volumes of all metric balls of radius r in a closed Riemannian manifold M~n do not exceed (r/c(n))~n, then the (n - 1)-dimensional Urysohn width of the manifold does not exceed r. In our version the assumption of Guth's theorem is relaxed to the assumption that for each x ∈ M~n there exists ρ(x) ∈ (0,r such that the volume of the metric ball B(x,ρ(x)) does not exceed (ρ(x)/c(n))~n, where one can take c(n) = 1/2n.
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