Given a closed strictly convex hypersurface M in the Euclidean space I?"+*, the Gauss map of M defines a homeomorphism between M and the unit sphere S". Therefore the Gauss-Kronecker curvature of M can be transplanted via the Gauss map to a function defined on S". If this function is denoted by K, then Minkowski observed that K must satisfy the equation where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a positive function K defined on S" satisfying the above integral conditions, can we find a closed strictly convex hypersurface whose curvature function is given by K? Minkowski solved the problem in the category of polyhedrons. Then AD Alexandrov and others solved the problem in general. However, this last solution does not provide any information about the regularity of the (unique) convex hypersurface even if we assume K is smooth. In the two
The Lp dual curvature measure was introduced by Lutwak,
Yang & Zhang in an attempt to unify the Lp Brunn–
Minkowski theory and the dual Brunn–Minkowski theory.
The characterization problem for Lp dual curvature measure,
called the Lp dual Minkowski problem, is a fundamental
problem in this unifying theory. The Lp dual Minkowski
problem contains the Lp Minkowski problem and the dual
Minkowski problem, two major problems in modern convex
geometry that remain open in general. In this paper, existence
results on the Lp dual Minkowski problem in the weak sense
will be provided. Moreover, existence and uniqueness of the
solution in the smooth category will also be demonstrated.
In this paper, we consider the L p dual Minkowski problem by geometric variational
method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence
of smooth solutions of the L p dual Minkowski problem when pq ≥ 0 and the
given data is even. If f ≡ 1, we show under some restrictions on p and q that the only
even, smooth, uniformly convex solution is the unit ball.
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 study of this paper consists of two aspects. One is characterizing the so-called circular cone convexity of f by exploiting the second-order differentiability of f L ; the other is introducing the concepts of determinant and trace associated with circular cone and establishing their basic inequalities. These results show the essential role played by the angle , which gives us a new insight when looking into properties about circular cone. MSC:26A27, 26B05, 26B35, 49J52, 90C33, 65K05.