We will discuss a correlation seen between the dark matter content and the ellipticity of elliptical galaxies. The analysis method for this investigation will be described and the origin of the correlation -whether it is physical or an observational/methodological bias- will be discussed. If of physical origin, the correlation found would imply that at equal luminosities, rounder medium-size elliptical galaxies appear to contain less dark matter than flatter elliptical galaxies. This would be puzzling in the context of the conventional model of cosmological structure formation. We will conclude by discussing a possible scenario explaining such correlation.