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