On non-compact Riemannian manifolds, we construct distance-like functions with derivatives controlled up to some order k assuming bounds on the growth of curvature ? 2 and decay injectivity radius. This construction extends previously known results in various directions, permitting obtain consequences which are (in a sense) sharp. As first main application, give refined conditions guaranteeing ...

Abstract We design a nonadaptive algorithm that, given oracle access to function which is ‐far from monotone, makes poly queries and returns an estimate with high probability, ‐approximation the distance of monotonicity. The analysis our relies on improvement directed isoperimetric inequality Khot, Minzer, Safra ( SIAM J. Comput. , 2018). Furthermore, we rule out ‐query that approximates monoto...