Hidden constant in eps-sample size computation Given a range space $(X,R)$ with VC-Dimension $\le d$, we can create an $\varepsilon$-sample with probability at least $1-\delta$ by sampling $ O\left(\frac{1}{\varepsilon^2}\left(d+\log\frac{1}{\delta}\right)\right)$ points with replacement from $X$.

Löffler and Phillips [1] showed experimentally that the constant hidden in the Big-Oh notation is at most 0.5. 
I was wondering whether there is any known theoretical, rather than experimental, upper bound to this constant. 

Thanks in advance for any reference

[1] Löffler, M and Phillips, J.M. "Shape fitting on point sets with probability distributions". ESA'09
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