I don't understand the first sentence in the second paragraph. What if the smallest is not extreme? If you are talking about unit spheres, then order the sphere by height and you get one-side kissing number plus one as upper bound. If you allow the spheres to have different radii, then order the spheres by size and you get the kissing number plus one as the upper bound. How can you order them by two criterions at the same time?

@cantwell, thank you for response. Your halved-cube example is wonderful, and was one of the motivation behind this answer. Actually, his problem can be regarded as the "opposite" of the Borsuk conjecture, which is very interesting.

Guys, I don't understand it. I just posted a better lower bound in another answer mathoverflow.net/a/195846/20595, then this answer got two votes ... what is happening?

Well, I must make a comment (confession) then: I believe I made some minor mistakes in the papers, not careful enough in the orientation. I will fix them in future papers.

There's a "separation function" due to Boyd, describing how close two spheres are or how deep they intersect. The inner product is exactly the negation of separation.