A very celebrated result on a related subject is the following, which was a very major advance in the Busemann-Petty problem (don't worry, the math review has all you need to know). EDIT by popular demand, the review can be seen here: http://dl.dropbox.com/u/5188175/BallReview.pdf
@incollection {MR950983,
AUTHOR = {Ball, Keith},
TITLE = {Some remarks on the geometry of convex sets},
BOOKTITLE = {Geometric aspects of functional analysis (1986/87)},
SERIES = {Lecture Notes in Math.},
VOLUME = {1317},
PAGES = {224--231},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1988},
MRCLASS = {52A40},
MRNUMBER = {950983 (89h:52009)},
MRREVIEWER = {G. D. Chakerian},
DOI = {10.1007/BFb0081743},
URL = {http://dx.doi.org/10.1007/BFb0081743},
}
