What is the volume of a \delta-ball in the orthogonal group O(n)? Is there a (simple) lower bound? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:26:21Z http://mathoverflow.net/feeds/question/2084 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2084/what-is-the-volume-of-a-delta-ball-in-the-orthogonal-group-on-is-there-a-sim What is the volume of a \delta-ball in the orthogonal group O(n)? Is there a (simple) lower bound? Skippy 2009-10-23T12:59:22Z 2011-06-10T18:41:37Z <p>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.</p> http://mathoverflow.net/questions/2084/what-is-the-volume-of-a-delta-ball-in-the-orthogonal-group-on-is-there-a-sim/2168#2168 Answer by David Bar Moshe for What is the volume of a \delta-ball in the orthogonal group O(n)? Is there a (simple) lower bound? David Bar Moshe 2009-10-23T19:49:47Z 2011-06-10T18:41:37Z <p>The volume of the delta-ball of the special orthogonal group can be computed exactly by applying the Weyl integration formula: (Without loss of generality, we assume that the delta-ball is around the unit group element).</p> <p>a. One notices (Again due to the invariance under the Haar measure) that the characteristic function of the delta ball is a class function. Thus upon the application of the Weyl integration formula we are left only with the radial part on the eigenvalues which is a $\lfloor N/2\rfloor$-dimensional integral for $\mathrm{SO}(N)$. Here, the radial integral is described explicitely.</p> <p>b. The eigenvalues of an orthogonal matrix of dimension $N=2m+1$ are $1$ and $m$ pairs $\exp(i \phi_ m)$ and $\exp(-i \phi_ m)$, $0\leq\phi_ 1 \leq\ldots\leq\phi_m \leq\pi$. In the case of even dimensions, the unit eigenvalue is absent.</p> <p>c. The delta-ball condition on the eigenvalues becomes:</p> <p>$$|\exp(i\phi_k)-1|\leq\delta ,$$ which implies: $$\phi_k\leq 2 \arcsin\sqrt{\delta/2}.$$</p> <p>d. Applying the Weyl integration formula, we obtain for the odd case $\mathrm{SO}(2m+1)$:</p> <p>$$\mathrm{Vol}(\delta\mathrm{-ball}) = \frac{2^{m^2}}{\pi^m m!} \int_{\phi_1\leq\ldots\leq\phi_m \leq 2 \arcsin\sqrt{\delta/2}} \prod_{1\leq j &lt; k \leq m} (\cos\phi_k-\cos\phi_j)^2 \prod_l \sin^2(\phi_l) d\phi_1 \cdots d\phi_k.$$</p> <p>e. For the even dimensional case, the only changes are $2^{m^2}$ is replaced $2^{(m-1)^2}$ and the sine terms are absent.</p>