Related papers: On a generalization of Busemann's intersection ine…
We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…
We consider the problem of comparing the volumes of two star bodies in an even-dimensional euclidean space $\mathbb R^{2n} = \mathbb C^n$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes)…
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…
One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem,…
Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is…
Busemann's intersection inequality asserts that the only maximizers of the integral $\int_{S^{n-1}} |K\cap\xi^\perp|^n d\xi$ among all convex bodies of a fixed volume in $\mathbb R^n$ are centered ellipsoids. We study this question in the…
The intersection body $IK$ of a star-body $K$ in $\mathbb{R}^n$ was introduced by E. Lutwak following the work of H. Busemann, and plays a central role in the dual Brunn-Minkowski theory. We show that when $n \geq 3$, $I^2 K = c K$ iff $K$…
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first…
We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections…
Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is…
We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann-Petty…
We investigate the intersections of balls of radius $r$, called $r$-ball bodies, in Euclidean $d$-space. An $r$-lense (resp., $r$-spindle) is the intersection of two balls of radius $r$ (resp., balls of radius $r$ containing a given pair of…
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…
We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain higher order…
The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose…
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…
The generalized Busemann-Petty problem asks whether centrally-symmetric convex bodies having larger volume of all m-dimensional sections necessarily have larger volume. When m>3 this is known to be false, but the cases m=2,3 are still open.…
The famous Minkowski inequality provides a sharp lower bound for the mixed volume $V(K,M[n-1])$ of two convex bodies $K,M\subset\mathbb{R}^n$ in terms of powers of the volumes of the individual bodies $K$ and $M$. The special case where $K$…
We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty…
In this paper, we consider the concept of $C$-star body in a fixed pointed closed convex cone $C$ and study the dual mixed volume for $C$-star bodies. For $C$-star bodies, we establish the corresponding dual Brunn-Minkowski inequality, the…