Isomorphic properties of Intersection bodies

We study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a convex body, when it exists and it is convex, with the Euclidean ball, is bounded by a constant depending only on k, generalizing a well-known result of Hensley and Borell. We conclude by giving some volumetric estimates for k-intersection bodies

Complex Intersection Bodies

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures and the corresponding hyperplane inequality for measures of complex intersection bodies

Modified Paouris inequality

The Paouris inequality gives the large deviation estimate for Euclidean norms of log-concave vectors. We present a modified version of it and show how the new inequality may be applied to derive tail estimates of l_r-norms and suprema of norms of coordinate projections of isotropic log-concave vectors.Comment: 14 page

Measure comparison and distance inequalities for convex bodies

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of $L_p$

On the equivalence of modes of convergence for log-concave measures

An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note, we explore the extent to which different notions of distance between probability measures are comparable for log-concave distributions. Our results imply that weak convergence of isotropic log-concave distributions is equivalent to convergence in total variation, and is further equivalent to convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note

Remarks on the KLS conjecture and Hardy-type inequalities

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann Poincar\'e constant on $\Omega$ to that of the cone and Lebesgue measures on $\partial \Omega$; these may be bounded via the curvature of $\partial \Omega$. A second reduction is obtained to the class of harmonic functions on $\Omega$. We also study the relation between the Poincar\'e constant of a log-concave measure $\mu$ and its associated K. Ball body $K_\mu$. In particular, we obtain a simple proof of a conjecture of Kannan--Lov\'asz--Simonovits for unit-balls of $\ell^n_p$, originally due to Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in final form in GAFA seminar note

A hypoid gear pair tribo-dynamic model taking into account the rheological behaviour of fully formulated gear lubricants

A fully coupled tribo-dynamic model, capable of predicting the inefficiency and dynamic response of automotive differential hypoid gear pairs, is presented in this study. A gear dynamics solver is coupled with an analytical friction solver, which calculates the viscous shear, as well as the boundary conjunctional friction force. The time varying geometry and contact characteristics of the hypoid gear pair are taken into account by using realistic data available in the literature. The rheological models employed cover a range of two different behaviors: Newtonian and non-Newtonian Eyring (shear thinning). The Chittenden-Dowson equation is used to calculate the central film thickness of the elasto-hydrodynamic teeth conjunctions. The boundary friction force is calculated using the Greenwood & Tripp model. Finally, the actual surface topography of a run-in hypoid gear is obtained using a stylus profilometer. The results indicate an overestimation of the viscous friction by the Newtonian model, as opposed with the non-Newtonian model, mainly due to shear thinning effects. Comparative studies are performed for different operating conditions, namely near or away from resonance, as well as for conditions corresponding to a non-linear sub-harmonic resonance. The frictional damping effect on the dynamic transmission error, which is an indication of the NVH response of the gear pair, is also examin

Isothermal Elastohydrodynamic Lubrication Analysis of Heavily Loaded Hypoid Gear Pairs

A numerical model able to predict the pressure distribution and the film thickness in heavily loaded elliptical EHL contacts is developed and presented in this study. The operating conditions, such as the contact load and the velocities of the mating surfaces, are representative of the corresponding conditions present in automotive differential hypoid gear pair units. The EHL solver presented is able to predict the minimum and central film thickness of the lubricating oil as well as the pressure distribution assuming isothermal and Newtonian conditions. Results are presented for a full quasi-static meshing cycle. A comparison between the numerically calculated values of the central and the minimum film thickness is performed against the corresponding values produced using the Chittenden-Dowson formula. A very good agreement is observed between the values of the central film thickness. However, it is shown that the minimum film thickness values using the Chittenden-Dowson formula can deviate up to 40% compared with the corresponding values which are calculated numerically

Film Thickness Investigation in Heavily Loaded Hypoid Gear Pair Elastohydrodynamic Conjunctions

Introduction: Hypoid gear pairs are some of the most highly loaded components of the differential unit in modern automobiles. Prediction of wear rate and generated friction require determination of lubricant film thickness. However, only very few investigations have addressed the issue of thin elastohydrodynamic films in hypoid gear pairs. The main reason for dearth of analysis in this regard has been the need for accurate determination of transient contact geometry and kinematics of interacting surfaces throughout a typical meshing cycle. Furthermore, combined gear dynamics and lubrication analysis of any pairs of simultaneous meshing teeth pairs is required. Simon [1] was among the first to deal with these issues. He used Tooth Contact Analysis (TCA) in order to calculate the instantaneous contact geometry and load for any teeth pair during their meshing cycle. However, in his study, the load carried by the hypoid pair was quite low, making the application of the results limited and not entirely suitable for real life operating conditions of typical hypoid gear pairs of vehicular differentials, which is of interest in the current paper. Xu and Kahraman [2] performed numerical prediction of power losses and consequently the film thickness for highly loaded hypoid gear pairs. However, in their study only the one-dimensional Reynolds equation was employed. Consequently, the effect of lubricant side leakage in the passage through the contact was ignored. A more recent study by Mohammadpour et al. [3] employed realistic gear geometry data (through the use of TCA) for calculation of film thickness time history through mesh. The two-dimensional Reynolds equation, accounting for the side leakage of the lubricant, was solved numerically. It was shown that the side leakage component of the entraining velocity can significantly influence the film thickness. With regard to hypoid gear dynamics, several studies should be mentioned. Wang and Lim [4] studied the dynamic response of hypoid gear pairs under the influence of time varying meshing stiffness. Yang and Lim [5] created a model able to predict the dynamic response of a hypoid gear pair by taking into account the lateral translations of their shafts due to the compliance of the supporting bearings. Karagiannis et al. [6-7] studied the dynamics of automotive differential hypoid gear pairs by taking into account the velocity dependent resistive torque at the gear caused by aerodynamic drag and tyre-road rolling resistance. The study integrated the gear dynamics with the generated viscous and boundary conjunctional friction