Is there a lower bound on the volume of a delta-ball in the orthogonal group O(n) of the type f(n) * delta^{n(n-1)/2}? For which f(n)? How can it be proven?

n(n-1)/2 is the number of degrees of freedom in the orthogonal group.

The volume in the orthogonal group is measured by the Haar measure, which is the up to scaling unique measure that is invariant under the group operation. I consider the usual metric that is induced by the spectral norm |M| = max |Mx| where x ranges over all vectors of length 1 and the vector norm is the Euclidean one. A delta-ball is the set of all orthogonal matrices that have distance less or equal delta to a fixed matrix M. Because of the invariance of the Haar measure, for a fixed delta, all delta-balls have the same volume.