Recently, the duals of Federer’s curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang & Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0,n). The dual Minkowski problem includes the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever q < 0, including both existence and uniqueness, is presented.
The relationship between Lp affine surface area and curvature measures is investigated. As a result, a new representation of the existing notion of Lp affine surface area depending only on curvature measures is derived. Direct proofs of the equivalence between this new representation and those previously known are provided. The proofs show that the new representation is, in a sense, “polar” to that of Lutwak’s and “dual” to that of Schutt & Werner’s.
For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.
It is proved that the classical Laplace transform is a continuous valuation which is positively GL(n) covariant and logarithmic translation covariant. Conversely, these properties turn out to be sucient to characterize this transform.
Petty proved that a convex body in R^n has the minimal surface area amongst its SL(n)images, if, and only if, its surface area measure is isotropic. By introducing a new notion of minimal Orlicz surface area, we generalize this result to the Orlicz setting. The analog of Ball’s reverse isoperimetric inequality is established.
The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that
of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy-
Kubota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established
for this new Orlicz mixed quermassintegrals.
A unified approach used to generalize classical Brunn-Minkowski
type inequalities to Lp Brunn-Minkowski type inequalities, called the Lp trans-
ference principle, is refined in this paper. As illustrations of the effectiveness
and practicability of this method, several new Lp Brunn-Minkowski type in-
equalities concerning the mixed volume, moment of inertia, quermassintegral,
projection body and capacity are established.
By using the method of mixed volumes, we give sharp bounds for inclusion measures of convex bodies in
n-dimensional Euclidean space. In the special cases where the random convex body is the unit ball or when
n = 3, neater and simpler bounds are obtained. All the associated inequalities proved are new isoperimetrictype
The Orlicz–Legendre ellipsoids, which are in the framework of emerging
dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in
some sense dual to the recently found Orlicz–John ellipsoids, and have largely generalized
the classical Legendre ellipsoid of inertia. Several new affine isoperimetric
inequalities are established. The connection between the characterization of Orlicz–
Legendre ellipsoids and isotropy of measures is demonstrated.
The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids. The analog of Ball’s volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.
Convex bodies with identical John and LYZ ellipsoids are characterized. This solves an
important problem from convex geometry posed by G. Zhang. As applications, several
sharp affine isoperimetric inequalities are established.
Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in R^n containing
the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine
invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the
conjecture in R^2 and R^3. Some new sharp inequalities characterizing parallelotopes in Rn are established.
Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes
in Rn is provided.
In analogy with the classical Minkowski problem, necessary and sufficient
conditions are given to assure that a given measure on the unit sphere is the cone-volume
measure of the unit ball of a finite dimensional Banach space.