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

Complex LpL_p-Intersection Bodies

Interpolating between the classic notions of intersection and polar centroid bodies, (real) $L_p$-intersection bodies, for $-1<p<1$, play an important role in the dual $L_p$-Brunn--Minkowski theory. Inspired by the recent construction of complex centroid bodies, a complex version of $L_p$-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 $L_p$-intersection body of an $\mathbb{S}^1$-invariant convex body is pseudo-convex, if $-2<p<-1$ and convex, if $p\geq-1$. Moreover, intersection inequalities of Busemann--Petty type in the sense of Adamczak--Paouris--Pivovarov--Simanjuntak are deduced.Comment: 32 page

Estimates for measures of sections of convex bodies

A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these inequalities from stability in comparison problems for different generalizations of intersection bodies

Palaeolithic extinctions and the Taurid Complex

Intersection with the debris of a large (50-100 km) short-period comet during the Upper Palaeolithic provides a satisfactory explanation for the catastrophe of celestial origin which has been postulated to have occurred around 12900 BP, and which presaged a return to ice age conditions of duration ~1300 years. The Taurid Complex appears to be the debris of this erstwhile comet; it includes at least 19 of the brightest near-Earth objects. Sub-kilometre bodies in meteor streams may present the greatest regional impact hazard on timescales of human concern.Comment: 7 pages, 3 figures; accepted for Monthly Notices of the Royal Astronomical Society (definitive version will be available at www.blackwell-synergy.com

Jade: A Differentiable Physics Engine for Articulated Rigid Bodies with Intersection-Free Frictional Contact

We present Jade, a differentiable physics engine for articulated rigid bodies. Jade models contacts as the Linear Complementarity Problem (LCP). Compared to existing differentiable simulations, Jade offers features including intersection-free collision simulation and stable LCP solutions for multiple frictional contacts. We use continuous collision detection to detect the time of impact and adopt the backtracking strategy to prevent intersection between bodies with complex geometry shapes. We derive the gradient calculation to ensure the whole simulation process is differentiable under the backtracking mechanism. We modify the popular Dantzig algorithm to get valid solutions under multiple frictional contacts. We conduct extensive experiments to demonstrate the effectiveness of our differentiable physics simulation over a variety of contact-rich tasks

Problematizing social justice in health pedagogy and youth sport: intersectionality of race, ethnicity, and class

Social justice education recognizes the discrepancies in opportunities among disadvantaged groups in society. The purpose of the articles in this special topic on social justice is to (a) provide a critical reflection on issues of social justice within health pedagogy and youth sport of Black and ethnic-minority (BME) young people; (b) provide a framework for the importance of intersectionality research (mainly the intersection of social class, race, and ethnicity) in youth sport and health pedagogy for social justice; and (c) contextualize the complex intersection and interplay of social issues (i.e., race, ethnicity, social classes) and their influence in shaping physical culture among young people with a BME background. The article argues that there are several social identities in any given pedagogical terrain that need to be heard and legitimized to avoid neglect and “othering.” This article suggests that a resurgence of interest in theoretical frameworks such as intersectionality can provide an effective platform to legitimize “non-normative bodies” (diverse bodies) in health pedagogy and physical education and sport by voicing positionalities on agency and practice

The complex Busemann-Petty problem on sections of convex bodies

The complex Busemann-Petty problem asks whether origin symmetric convex bodies in \C^n with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$Comment: 18 page

An isomorphic version of the Busemann-Petty problem for arbitrary measures

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le \sqrt{n} \mu(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures

Mixed volume and an extension of intersection theory of divisors

Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of solutions in X of a system of equations f_1 = ... = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L_1,..., L_n] can be extended to the Grothendieck group of K(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein-Kushnirenko theorem from the Newton polytope theory.Comment: 31 pages. To appear in Moscow Mathematical Journa