Wide short geodesic loops on closed Riemannian manifolds.

It is not known whether or not the length of the shortest periodic geodesic on a closed Riemannian manifold M n can be majorized by c (n) v o l 1 n , or c ̃ (n) d , where n is the dimension of M n , v o l denotes the volume of M n , and d denotes its diameter. In this paper, we will prove that for each > 0 one can find such estimates for the length of a geodesic loop with angle between π − and π with an explicit constant that depends both on n and . That is, let > 0 , and let a = ⌈ 1 sin ( 2) ⌉ + 1. We will prove that there exists a "wide" (i.e. with an angle that is wider than π − ) geodesic loop on M n of length at most 2 n ! a n d. We will also show that there exists a "wide" geodesic loop of length at most 2 (n + 1) ! 2 a (n + 1) 3 F i l l R a d ≤ 2 ⋅ n (n + 1) ! 2 a (n + 1) 3 v o l 1 n . Here F i l l R a d is the Filling Radius of M n . [ABSTRACT FROM AUTHOR]