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Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under isometry and satisfy $S(g) \leq S(g')$ if $g \leq g'$, where $g \leq g'$ means $g(v,v) \leq g'(v,v)$ for all $v \in TM$.

For an open Riemannian manifold $(M, g)$ and for a compact set $K \subset M$ define $$S_M(g, K) = \inf \{||f|| : f \in C^\infty_c(M) \mbox{ such that } |\nabla f(x)|_g \geq 1 \mbox{ for all } x\in K\}$$ where $||f|| = \max_M f - \min_M f$, and then define $$S_M(g) = \sup_{K\subset M} S_M(g, K).$$ One can prove that this is an size invariant.

This invariant appears in Kei Irie's paper 'Displacement energy of unit cotangent bundles' in Section 2, http://arxiv.org/abs/1106.2199

Irie proves that $S_M(g) \leq c_n r(M, g)$ where $c_n$ is a dimensional constant and $r(M,g)$ is the inner radius. Irie also lower bounds $S_M(g)$ in terms of the displacement energy of the unit disk cotangent bundle $D^*_g M$ in the symplectic manifold $T^*M$ (it is important here that $M$ is open).T^*M$. My question is if this invariant appears elsewhere in the literature or is related to other known invariants? 5 added 202 characters in body Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian metrics on a smooth manifold$M$. These are functions$S: Metrics(M) \to \mathbb{R}$that are invariant under isometry and satisfy$S(g) \leq S(g')$if$g \leq g'$, where$g \leq g'$means$g(v,v) \leq g'(v,v)$for all$v \in TM$. For an open Riemannian manifold$(M, g)$and for a compact set$K \subset M$define $$S_M(g, K) = \inf \{||f|| : f \in C^\infty_c(M) \mbox{ such that } |\nabla f(x)|_g \geq 1 \mbox{ for all } x\in K\}$$ where$||f|| = \max_M f - \min_M f$, and then define $$S_M(g) = \sup_{K\subset M} S_M(g, K).$$ One can prove that this is an size invariant. This invariant appears in Kei Irie's paper 'Displacement energy of unit cotangent bundles' in Section 2, http://arxiv.org/abs/1106.2199 Irie proves that$S_M(g) \leq c_n r(M, g)$where$c_n$is a dimensional constant and$r(M,g)$is the inner radius. Irie also lower bounds$S_M(g)$in terms of the displacement energy of the unit disk cotangent bundle$D^*_g M$in the symplectic manifold$T^*M$(it is important here that$M\$ is open).

My question is if this invariant appears elsewhere in the literature or is related to other known invariants?