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 Mn can be majorized by c(n)vol 1 n, or c(n)d, where n is the dimension of Mn, vol denotes the volume of Mn, 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 pi - and pi with an explicit constant that depends both on n and .That is, let > 0, and let a = left ceiling 1 sin( 2 ) right ceiling + 1. We will prove that there exists a "wide" (i.e. with an angle that is wider than pi - ) geodesic loop on Mn of length at most 2n!and. We will also show that there exists a "wide" geodesic loop of length at most 2(n+1)!2a(n+1)3F illRad <= 2<middle dot>n(n+1)!2a(n+1)3vol1n Here FillRad is the Filling Radius of Mn.