Related papers: Mixed volumes of normal complexes

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 $K$…

The geometry of the dual amplituhedron is generally described in reference to a particular triangulation. A given triangulation manifests only certain aspects of the underlying space while obscuring others, therefore understanding this…

Here a new condition for the geometry of Banach spaces is introduced and the operator--valued Fourier multiplier theorems in weighted Besov spaces are obtained. Particularly, connections between the geometry of Banach spaces and…

Let a three-dimensional hyperbolic polyhedron $\mathcal P$ have finite volume $\mathrm{vol}(\mathcal P)$ and a finite number of vertices $\mathrm{ver}(\mathcal P)$. We call its normalized volume the quantity $\omega(\mathcal P) =…

Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the…

We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data…

We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex…

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the…

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal…

We study gravitationally bound, spherically symmetric equilibrium configurations consisting of ordinary (neutron-star) matter and of a phantom/ghost scalar field which provides the nontrivial topology in the system. For such mixed…

It has been shown by Soprunov that the normalized mixed volume (minus one) of an $n$-tuple of $n$-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined…

Describing the equality conditions of the Alexandrov--Fenchel inequality has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial…

A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset, determine whether there exists a polytope whose face lattice is the poset. In 1990s, Kapranov defined a poset as a hybrid between…

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric…

Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable…

In this paper, we give a rigorous proof for the expression of the angle between adjacent sides in the skew polygons appearing at rational times in the evolution of regular polygons of $M$ sides under the vortex filament equation. The proof…

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…

For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete K\"ahler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply…

We verify the Rota-Heron-Welsh conjecture for matroids realizable as c-arrangements: the coefficients of the characteristic polynomial of the associated matroid are log-concave. This family of matroids strictly contains that of complex…

A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that…