The Morse landscape of a Riemannian disk (Q685891)

We study upper bounds on the length functional along contractions of loops in Riemannian disks of bounded diameter and circumference. By constructing metrics adapted to imbedded trees of increasing complexity, we reduce the nonexistence of such upper bounds to the study of a topological invariant of imbedded finite trees. This invariant is related to the complexity of the binary representation of integers. It is also related to lower bounds on the number of points in level sets of a real- valued function on the tree. Our construction answers in the negative a question of Gromov, which was motivated by the study of the word metric on finitely generated groups.