Lp Intersection Bodies Research Articles

Complex Lp-intersection bodies

Interpolating between the classical notions of intersection and polar centroid bodies, (real) Lp-intersection bodies, for −1<p<1, play an important role in the dual Lp-Brunn–Minkowski theory. Inspired by the recent construction of complex centroid bodies, a complex version of Lp-intersection bodies, with range extended to p>−2, is introduced, interpolating between complex intersection and polar complex centroid bodies. It is shown that the complex Lp-intersection body of an S1-invariant convex body is pseudo-convex, if −2<p<−1 and convex, if p≥−1. Moreover, intersection inequalities of Busemann–Petty type in the sense of Adamczak–Paouris–Pivovarov–Simanjuntak are deduced.

Inequalities on the (p,q)-mixed volume involving Lp centroid bodies and Lp intersection bodies

In this paper, applying for the Minkowski?s and H?lder?s integral inequalities, we obtain four theorems about the (p, q)-mixed volume involving the Lp centroid bodies and the Lp intersection bodies, respectively. The former two theorems reveal the convexity of the functionals related to the (p, q)-mixed volume, in terms of the dual Blaschke addition introduced in [Journal of Geometric Analysis, 30 (2020) 3026-3034], and the latter two theorems expose the monotonicity of the other functionals related to the (p, q)-mixed volume.

Lp-dual affine surface areas for the general Lp-intersection bodies

For 0 &lt; p &lt; 1, the notions of symmetric and asymmetric Lp-intersection bodies were introduced by Haberl and Ludwig. Recently, Wang and Li defined the general Lp-intersection bodies. In this paper, associated with the Lp-dual affine surface areas, we give the extremum values of the general Lp-intersection bodies. Moreover, a Brunn-Minkowski type inequality and a monotone inequality for the Lp-dual affine surface area version of general Lp-intersection bodies are established, respectively.

Generalized Intersection Bodies with Parameter

AIn 2005, the classical intersection bodies and Lp intersection bodies were extended. Afterwards, the concept of general Lp intersection bodies and the generalized intersection bodies was introduced. In this paper, we define the generalized bodies with parameter. Besides, we establish the extremal values for volume, Brunn-Minkowski type inequality for radial combination and Lp harmonic Blaschke combination of this notion.

Lp -dual geominimal surface areas for the general Lp-intersection bodies

Lp -dual geominimal surface areas for the general Lp-intersection bodies

Busemann-Petty Problems for Quasi Lp Intersection Bodies

We investigate the Lp dual geominimal surface area and volume forms of Busemann-Petty problems for the quasi Lp intersection bodies and establish some new geometric inequalities. Our results provide a significant complement to the researches on Busemann-Petty problems for intersection bodies.

A type of Busemann-Petty problems for general Lp-intersection bodies

Recently, the notion of general (containing symmetric and asymmetric) Lp-intersection bodies was given. In this article, by the Lp-dual mixed volumes and the general Lp-dual Blaschke bodies, we study the Lp-dual affine surface area forms of the Busemann-Petty problems for general Lp-intersection bodies. Our works belong to a new and rapidly evolving asymmetric Lp-Brunn-Minkowski theory.

Busemann-Petty problems for general L p -intersection bodies

For 0 < p < 1, Haberl and Ludwig defined the notions of symmetric Lp-intersection body and nonsymmetric Lp-intersection body. In this paper, we introduce the general Lp-intersection bodies. Furthermore, the Busemann-Petty problems for the general Lp-intersection bodies are shown.

L p -dual affine surface area forms of Busemann–Petty type problems

Associated with the notion of Lp-intersection body which was defined by Haberl, we research Lp-dual affine surface area forms of Busemann–Petty type problems.

L p -mixed intersection bodies and star duality

The paper extends the two notions of the dual mixed volumes and Lp-intersection body to q-dual mixed volumes and Lp-mixed intersection body, respectively. Inequalities for the star dual of Lp-mixed intersection bodies are established.

A characterization of Lp intersection bodies

All GL(n) covariant Lp radial valuations on convex polytopes are classified for every p > 0. It is shown that for 0 < p < 1 there is a unique non-trivial such valuation with centrally symmetric images. This establishes a characterization of Lp intersection bodies. 2000 AMS subject classification: 52A20 (52B11, 52B45)