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A note on the richness of convex hulls of VC classes

   
@article{ECP1097,
	author = {Gábor Lugosi and Shahar Mendelson and Vladimir Koltchinskii},
	title = {A note on the richness of convex hulls of VC classes},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {8},
	year = {2003},
	keywords = {},
	abstract = {We prove the existence of a class $A$ of subsets of $\mathbb{R}^d$ of VC dimension 1 such that the symmetric convex hull $F$ of the class of characteristic functions of sets in $A$ is rich in the following sense. For any absolutely continuous probability measure $\mu$ on $\mathbb{R}^d$, measurable set $B$ and $\varepsilon > 0$, there exists a function $f$ in $F$ such that the measure of the symmetric difference of $B$ and the set where $f$ is positive is less than $\varepsilon$. The question was motivated by the investigation of the theoretical properties of certain algorithms in machine learning.},
	pages = {no. 18, 167-169},
	issn = {1083-589X},
	doi = {10.1214/ECP.v8-1097},    
        url = {http://ecp.ejpecp.org/article/view/1097}}