Gromov’s systolic inequality asserts that the length,
, of
the shortest noncontractible curve in a closed essential Riemannian manifold
does not
exceed
for
some constant
.
(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
. (The best previously
known upper bound for
was exponential in .)
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
in a closed
Riemannian manifold
do not exceed
, then
the
–dimensional
Urysohn width of the manifold does not exceed
. In our
version the assumption of Guth’s theorem is relaxed to the assumption that for each
there exists
such that the volume
of the metric ball
does not exceed
,
where one can take
.
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