We consider the problem of classifying the convex bodies in the 3-dimensional space depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner/outer parallel bodies. It turns out that this problem is closely related to some behavior of the roots of the 3-dimensional Steiner polynomial.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-8,
author = {Eugenia Saor\'\i n},
title = {On Hadwiger's problem on inner parallel bodies},
journal = {Banach Center Publications},
volume = {86},
year = {2009},
pages = {121-129},
zbl = {1167.52007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-8}
}
Eugenia Saorín. On Hadwiger's problem on inner parallel bodies. Banach Center Publications, Tome 86 (2009) pp. 121-129. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-8/