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  "body": "\n## Statement\n\nLet $n \\ge 1$ be an integer.\nLet $M$ be a closed connected manifold of dimension $n$ with fundamental group $\\pi$.\nLet $f \\colon M \\to K(\\pi, 1)$ be a map that induces an isomorphism of fundamental groups.\nLet $A = \\Z$ if $M$ is orientable and $A = \\Z/2\\Z$ otherwise.\nCall $M$ *essential* if $f$ sends the fundamental class of $M$ to a nonzero class of $H_n(K(\\pi, 1); A)$.\n\nThere is a constant $C_n \u003e 0$, depending only on $n$, with the following property.\nFor every essential $M$ of dimension $n$ and every [[Riemannian metric|Q61VFHCH2J]] $g$ on $M$,\n$$\\operatorname{sys}(M, g)^n \\le C_n \\operatorname{vol}(M, g),$$\nwhere $\\operatorname{sys}$ is the [[systole|PZCJV40M4K]].\n\n## Remarks\n\nGromov's proof goes through the filling radius.\nOne embeds $M$ isometrically into the Banach space $L^\\infty(M)$ by the Kuratowski embedding.\nFor essential $M$ the filling radius is at least $\\operatorname{sys}(M, g)/6$.\nOn the other hand, a filling inequality bounds the filling radius by a constant times $\\operatorname{vol}(M, g)^{1/n}$.\n\nThe tori $T^n$, the real projective spaces $\\R P^n$ and all closed aspherical manifolds are essential.\nFor $T^2$ the inequality is a weak form of [[Loewner's torus inequality|1Y9X1YW6GH]].\n\nThe hypothesis that $M$ is essential cannot be dropped.\nOn $S^2 \\times S^1$ take the product of a round metric of small area $\\varepsilon$ on $S^2$ and a circle of length $1$.\nEvery noncontractible loop winds around the circle factor, so the systole is $1$, while the volume $\\varepsilon$ tends to $0$.\n\n## Explanations\n\n### L1 human\n\nFor many manifolds, the shortest loop that cannot be shrunk to a point controls the volume.\nIf every such loop has length at least $L$, then the volume is at least a fixed multiple of $L^n$.\nThe theorem gives this for a large class of manifolds, the essential ones, in every dimension.\n\n### L2 human\n\nThis is the higher-dimensional extension of Loewner's and Pu's inequalities in [[systolic geometry|PZCJV40M4K]].\nThe constant $C_n$ from the proof is not optimal, and optimal constants are unknown in most cases.\nThe notion of essential manifold is topological: it asks that the fundamental class survive in the homology of the classifying space of the fundamental group.\n",
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