A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a ﬁnite dimensional variational principle.
The dual Loomis-Whitney inequality provides the sharp lower bound
for the volume of a convex body in terms of its $(n-1)$-dimensional
coordinate sections. In this paper, some reverse forms of the dual
Loomis-Whitney inequality are obtained. In particular, we show that
the best universal DLW-constant for origin-symmetric planar convex
bodies is $1$.
Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two
new sine ellipsoids are introduced in this paper. These four
ellipsoids are closely related in the Pythagorean relation and
duality. Several volume inequalities and the valuation
properties are obtained for two new