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In this paper, the ( p , q ) -mixed geominimal surface area and the ( p , q ) -mixed affine surface area are introduced. They are extensions of the L p geominimal surface area and the L p affine surface area, respectively. Affine invariance under unimodular affine transformations and continuity for the ( p , q ) -mixed geominimal surface area and the ( p , q ) -mixed affine surface area are discussed. The existence, the uniqueness and the uniform boundness of the ( p , q ) -mixed Petty bodies are obtained. Affine isoperimetric inequalities for the ( p , q ) -mixed geominimal surface area and the ( p , q ) -mixed affine surface area are established.

This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in ℝ2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.

The aims of this paper are to develop the LYZ ellipsoid and Petty projection body for log-concave functions, which correspond to the LYZ ellipsoid and Petty projection body for convex bodies when restricted to the subclass of characteristic functions. Moreover, the continuous, SL ( n ) contravariant valuation on a subclass of log-concave functions is classified.

The maximum platoon size is a critical parameter in connected and automated vehicle (CAV) platoon configuration. However, the effect of platoon size on the transportation system has not been well-studied. This paper unveils the effect of maximum CAV platoon size in terms of road capacity and traffic flow stability. Specifically, the analytical formulations of the capacity and flow stability are developed considering the maximum platoon size. Simulations are conducted to verify the developed theoretical models. For capacity analysis, both the analytical and simulation results indicate that a larger maximum platoon size can help increase the capacity. However, the increment becomes smaller with the increase of maximum platoon size. For flow stability analysis, the theoretical analysis and microscopic simulation show that smaller maximum platoon size leads to greater traffic flow stabilization. In addition, analysis shows that improvements in capacity and traffic stability are more profound when CAV penetration and platooning intensity are high. Ministry of Education (MOE) This study is supported by Singapore Ministry of Education Academic Research Fund Tier 2 MOE2017-T2-1-029.

Lutwak et al. (2010)established the Orlicz centroid inequality for convex bodies andconjectured that their Orlicz centroid inequality could be extendedto star bodies. Zhu (2012) confirmed the conjectured Lutwak, Yang and Zhang (LYZ) Orlicz centroidinequality and solved the equality condition for the case that$\phi$ is strictly convex. Without the condition that $\phi$ isstrictly convex, this paper studies the equality condition of theconjectured LYZ Orlicz centroid inequality for star bodies.

In this paper, the uniqueness of the dual Minkowski problem for the dual area measure is established via the dual Minkowski inequality and the dual log-Minkowski inequality. For real q > n − 1 , it is proved that the weak convergence of the q-th dual area measure implies the convergence of the corresponding convex bodies in the Hausdorff metric and that the solution to the dual Minkowski problem is continuous with respect to q.

By using the moving frame method, the authors obtain a kind of asymmetric kinematic formulas for the total mean curvatures of hypersurfaces in the n-dimensional Euclidean space.

We first investigate the translative containment measure for convex domain K0 to contain, or to be contained in, the homothetic copy of another convex domain K1, i.e., given two convex domains K0, K1 of areas A0, A1, respectively, in the Euclidean plane ℝ2, is there a translation T so that t(TK1) ⊂ K0 or t(TK1) ⊃ K0 for t > 0? Via the translative kinematic formulas of Poincare and Blaschke in integral geometry, we estimate the symmetric mixed isohomothetic deficit σ2(K0,K1) ≡ A012 − A0A1, where A01 is the mixed area of K0 and K1. We obtain a sufficient condition for K0 to contain, or to be contained in, t(TK1). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.

The notion of total mean curvature matrix of a submanifold in R n is defined. A kinematic integral formula for the total mean curvature matrix is proved.

investigate the symmetric mixed isoperimetric deficit Δ2( K 0, K 1) of domains K 0 and K 1 in the Euclidean plane R 2 via the known kinematic formulas of Poincare and Blaschke in integral geometry. Then we obtain symmetric mixed isoperimetric inequality and some Bonnesen-style symmetric mixed inequalities. One of those inequalities strengthens Kotlyars inequality. Finally, we obtain some reverse Bonnesen-style symmetric mixed inequalities that generalize the known Bottemas result.

This paper deals with L p geominimal surface area and its extension to L p mixed geominimal surface area. We give an integral formula of L p geominimal surface area by the p -Petty body and introduce the concept of L p mixed geominimal surface area which is a natural extension of L p geominimal surface area. Some inequalities, such as, analogues of Alexandrov–Fenchel inequalities, Blaschke–Santalo inequalities, and affine isoperimetric inequalities for L p mixed geominimal surface areas are obtained.

By Cheeger’s isoperimetric constants, some lower bounds and upper bounds of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda_{1}$\end{document}λ1, the first eigenvalue on a complete surface of constant curvature, are given. Some Bonnesen-style inequalities and reverse Bonnesen-style inequalities for the first eigenvalue are obtained. Those Bonnesen-style inequalities obtained are stronger than the well-known Osserman’s results and the upper bound is stronger than Osserman’s results (Osserman in Proceedings of the International Congress of Mathematicians, Helsinki, 1978).

We study the dual Orlicz mixed Quermassintegral. For arbitrary monotone continuous function \(\phi \), the dual Orlicz radial sum and dual Orlicz mixed Quermassintegral are introduced. Then the dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral are obtained. These inequalities are just the special cases of their \(L_p\) analogues (including cases \(-\infty

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.

In this paper, the Orlicz-affine surface area is introduced. Isoperimetric inequalities for this new afine surface area are established. Mathematics subject classification (2010): Primary 52A20; Secondary 53A15.

We obtain a sharp convex mixed Lorentz-Sobolev inequality, which is the functional version of an L p Minkowski inequality. The new sharp convex mixed Lorentz-Sobolev inequality implies the sharp convex Lorentz-Sobolev inequality of Ludwig, Xiao and Zhang.

About 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke–Santalo inequalities, J. Differ. Geom. 47 (1997), 1–16) introduced the notion of Lp moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26). In this paper, we establish new Brunn–Minkowski-type inequalities with respect to Blaschke Lp harmonic addition for the quermassintegrals and dual quermassintegrals of Lp moment bodies.

In this paper, the reverse Bonnesen style inequalities for con- vex domain in the Euclidean plane R 2 are investigated. The Minkowski mixed convex set of two convex sets K and L is studied and some new geo- metric inequalities are obtained. From these inequalities obtained, some isoperimetric deficit upper limits, that is, the reverse Bonnesen style in- equalities for convex domain K are obtained. These isoperimetric deficit upper limits obtained are more fundamental than the known results of Bottema ((5)) and Pleijel ((22)).

The Wulff isoperimetric inequality is a natural extension of the classical isoperimetric inequality (Green and Osher in Asian J. Math. 3:659-676 1999). In this paper, we establish some Bonnesen-style Wulff isoperimetric inequalities and reverse Bonnesen-style Wulff isoperimetric inequalities. Those inequalities obtained are extensions of known Bonnesen-style inequalities and reverse Bonnesen-style inequalities.

We first estimate the containment measure of a convex domain to contain in another in a surface $$\mathbb{X}_\varepsilon$$ of constant curvature e. Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in $$\mathbb{X}_\varepsilon$$ . Finally we strengthen the known Bonnesen isoperimetric inequality.

In this paper, the Orlicz centroid body, defined by E. Lutwak, D. Yang and G. Zhang, and the extrema of some affine invariant functionals involving the volume of the Orlicz centroid body are investigated. The reverse form of the Orlicz Busemann–Petty centroid inequalities is obtained in the two-dimensional case.

In this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

In this paper, we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface $$ \mathbb{X} $$ ∈ of constant curvature ∈, that is, an isoperimetric deficit upper bound of the convex domain in $$ \mathbb{X} $$ ∈ . The result is an analogue of the known Bottema’s result of 1933 in the Euclidean plane $$ \mathbb{E} $$ 2.

We investigate the containment measure of one domain to contain in another domain in a plane Xκ of constant curvature. We obtain some Bonnesen-type inequalities involving the area, length, radius of the inscribed and the circumscribed disc of a domain D in Xκ.

We present some L p affine isoperimetric inequalities for L p geominimal surface area. In particular, we obtain an analogue of Blaschke-Santalo inequality. We give an integral formula of L p geominimal surface area by the p-Petty body. Furthermore, we introduce the concept of L p mixed geominimal surface area which is a nature extension of L p geominimal surface area. We also extend Lutwak’s results for L p mixed geominimal surface area.

For convex domains K i (i = 0, 1) (compact convex sets with non-empty interiors) in the Euclidean plane R 2. Denote by A i and P i areas and circum-perimeters, respectively. The symmetric mixed isoperimetric deficit is \(\varDelta (K_{0},K_{1}):= P_{0}^{2}P_{1}^{2} - 16\pi ^{2}A_{0}A_{1}\). In this paper, we give some Bonnesen-style symmetric mixed inequalities, that is, inequalities of the form \(\varDelta (K_{0},K_{1}) \geq B_{K_{0},K_{1}}\), where \(B_{K_{0},K_{1}}\) is a non-negative invariant of geometric significance and vanishes if and only if both K 0 and K 1 are discs. We also obtain some reverse Bonnesen-style symmetric mixed inequalities. Those inequalities are natural generalizations of known geometric inequalities, such as the known classical isoperimetric inequality.

As an application of the analogue of C-S. Chen’s kinematic formula in the 3-dimensional space of constant curvature ϵ \epsilon , that is, Euclidean space R 3 {\mathbb {R}}^{3} , 3 3 -sphere S 3 S^{3} , hyperbolic space H 3 {\mathbb {H}}^{3} ( ϵ = 0 , + 1 , − 1 \epsilon =0,\,+1,\,-1 , respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space R 3 \mathbb {R}^{3} , or a 3 3 -sphere S 3 S^{3} or a hyperbolic space H 3 \mathbb {H}^{3} .

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are proved. Related inequalities, such as, Alexander-Fenchel type inequality, Santal\'{o} style inequality, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, we also study some properties and inequalities for the $i$-th mixed $L_p$ geominimal surface areas for two convex bodies.

In this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

Let $G$ be a Lie group and $H$ its subgroup, and let $M^p$, $N^q$ be two submanifolds of dimensions $p$, $q$, respectively, in the Riemannian homogeneous space $G/H$. We study the relationships between the second fundamental forms of $M^p \cap gN^q$ and the second fundamental forms of $M^p$, $N^q$ for $g \in G$. We find that the second fundamental form of $M^p \cap gN^q$ can be expressed by the curvature functions of $M^p$, $N^q$ and the “angle” between $M^p$ and $N^q$. All results achieved are the generalizations of known results of the classical differential geometry in $\mathrm{R}^3$.