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.

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

You can find it here - http://wenku.baidu.com/view/0514da4d852458fb770b56c6.html?from=related

Diameters are double normals!

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.

There is a proof (by Maxim Kazaryan) of my estimate for the case of embeddings, which use only square-function - http://www.mi.ras.ru/~kazarian/papers/homology05.pdf in the section 16. Unfortunately it is in Russian.