Thanks. I can almost see the connection. But I cannot yet prove it that the kissing number is exactly the quantity of interest (supposing I allow for ties). Sorry I am a bit slow today.

Oh I think I may be seeing the argument now. Please correct me if I am wrong. $S_l$ being small in this case, will mean that 100 occurs more than expected, and thus the probability $S_n$ being larger increases.

Hi, thanks for the answer. I am still a bit confused. "Now conditioning on S2 being small actually makes S1 larger." What is S2 and S1 here? n > l in the ineuqality, so if we are considering l = 1 and n = 2, then the conditioning should be on S1. It may be that I have not understood the argument.

Thanks for the reference. I will read this and get back.

Thanks, this makes sense. The standard property that you mention should follow from Jensen's.

dl.acm.org/citation.cfm?id=2745847 Lemma 10.2