Related papers: An analytical approach to the Rational Simplex Pro…

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…

Macdonald studied a discrete volume measure for a rational polytope $P$, called solid angle sum, that gives a natural discrete volume for $P$. We give a local formula for the codimension two quasi-coefficient of the solid angle sum of $P$.…

The cevians passing through a point in a simplex create a cevian simplex, which is divided by these cevians into smaller simplices. We consider the problem about the maximum of the ratio of the sum of the volumes of some of these smaller…

H. Busemann and C. M. Petty posed the following problem in 1956: If K and L are origin-symmetric convex bodies in R^n and for each hyperplane H through the origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H), does…

Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic…

Let $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$, where $\kappa=(k_1,\dots,k_n)$, be a stratum of (projectivized) $d$-differentials in genus $0$. We prove a recursive formula which relates the volume of…

In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the…

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is…

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces,…

Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits…

We provide sharp lower bounds for the simplicial volume of compact $3$-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of $3$-manifolds, including…

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four…

We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real provided that the data are real. Our arguments explore the connection between subspaces of codimension 2…

We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $\pi$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples…

Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The…

A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a…

A subequation on an open subset $X\subset \mathbb R^n$ is a subset $F$ of the space of $2$-jets on $X$ with certain properties. A smooth function is said to be $F$-subharmonic if all of its $2$-jets lie in $F$, and using the viscosity…

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in…

For any $n$-dimensional simplex in the Euclidean space $\mathbb{R}^n$ with $n\ge 4$, it is asked that if a continuous deformation preserves the volumes of all the codimension 2 faces, then is it necessarily a \emph{rigid} motion. While the…

This note is a comment to the paper by D.R.Heath-Brown and B.Z.Moroz (Math Proc. Camb. Phil. Soc. 125 (1999)). That paper concerns with the projective surface $S$ in $\mathbb{P}^{3}$ defined by the equation $x_{1}x_{2}x_{3}=x_{4}^{3}$. It…