I suggest to look at my paper - P. E. Pushkar', “Generalization of the Chekanov Theorem. Diameters of Immersed Manifolds and Wave Fronts” Local and global problems of singularity theory, Collection of papers dedicated to the 60th anniversary of academician Vladimir Igorevich Arnold, Tr. Mat. Inst. Steklova, 221, Nauka, Moscow, 1998, 289–304
Diameters are double normals! It was translated, hope you can find it. I also shoud have the trabslation somewhere..
In particular, there is an estimate in the paper - number of double normals of generic immersed submanifold $M^n$ of the Euclidean space is at least $(B^2-B)/2+nB/2$. Here $B$ is $\dim H_*(M,Z_2)$. This estimation is exact for product of spheres, oriented surfaces.