Non-intersection Bodies All of Whose Central Sections Are Intersection Bodies

Volume difference inequalities for the projection and intersection bodies

In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.

On Strict Inclusions in Hierarchies of Convex Bodies

Let Ik be the class of convex k-intersection bodies in R n (in the sense of Koldobsky) and Im k be the class of convex origin-symmetric bodies all of whose m-dimensional central sections are k-intersection bodies. We show that 1) Im k 6⊂ I m+1 k , k + 3 ≤ m < n, and 2) Il 6⊂ Ik, 1 ≤ k < l < n − 3.

Intersection Bodies and Generalized Cosine Transforms

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. We review some known facts and give them new proofs. The main focus is interrelation between generalized cosine transforms of different kinds and their application to investigation of certai...

Intersection graphs associated with semigroup acts

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...

The Brunn-Minkowski-Type Inequality