Hidden constant in eps-sample size computation
I discovered after this publication that there is older work by 
Dvoretzky-Kiefer-Wolfowitz in 1956 that looks specifically at this one-dimensional case for sampling.  There is also follow up by just Kiefer and Wolfowitz a year or two later that handles the (anchored) rectangle case.  

For these specific settings, there has been some much more refined work looking at the actual constants.  Here is a reference that might help: 


P. Massart, "The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality", 1990.


I don't know off-hand any other work on the constants other than the 1-d or "anchored" rectangle cases

Jeff