Spaces with almost Euclidean Dehn function

Abstract

We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are CAT(0). This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper CAT(0) in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being CAT(0) up to that scale.Date: November 9, 2018. 1991 Mathematics Subject Classification. 53C23, 20F65, 49Q05.

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

“…This result is sharp as the Dehn function of the Euclidean space R n is given by 1 4 r 2 , independently of A, see Example 4.4. The implications of a non-strict inequality in (1.3) have been investigated in [52].…”

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

“…This result is sharp as the Dehn function δ of the Euclidean space R n is given by δ(r) = 1 4π ⋅ r 2 independently of A, see Example 4.3. The implications of a non-strict inequality in (4) have been investigated in [Wen19].…”