In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness
of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric
is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the
standard sphere if we fix its geometric center in the ambient space. Finally, we discuss a Shi-Tam type of inequality for the
Schwarzschild manifold as an application of our findings.
A variational formula for the Lutwak affine surface areas j of convex bodies in Rn is
established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with
projection functions of convex bodies, we prove a sharp isoperimetric inequality for j ,
which opens up a new passage to attack the longstanding Lutwak conjecture in convex
Existence and uniqueness of the solution to the Lp Minkowski
problem for the electrostatic p-capacity are proved when p > 1
and 1 < p < n. These results are nonlinear extensions of the very
recent solution to the Lp Minkowski problem for p-capacity when
p = 1 and 1 < p < n by Colesanti et al. and Akman et al., and
the classical solution to the Minkowski problem for electrostatic
capacity when p = 1 and p = 2 by Jerison.