Consider a random set of points on the unit sphere in $mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$,
for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.
We prove some recent experimental observations of D. Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the one-parameter family of such polygons (t
hat exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.
In this paper, extending the works of Milena Radnovic and Serge Tabachnikov, we establish conditions for two different non-symmetric norms to define the same billiard reflection law.
Representing an atom by a solid sphere in $3$-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free ene
rgy. The morphometric approach [HRC13,RHK06] writes the latter as a linear combination of weight
The morphometric approach [HRC13,RHK06] writes the solvation free energy as a linear combination of weight
The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric
proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.
We study the Gromov waist in the sense of $t$-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromovs
original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no $t$-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromovs pancake argument to produce some estimates of $t$-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. For readers convenience, in one appendix of this paper we provide a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. In the other appendix, we provide a comparison of different variations of Gromovs pancake method.
We give a simple proof of T. Stehlings result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.
We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $mge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible higher-dimensional generali
zations and difficulties of extending our technique to equalizing more than one non-additive function.
We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
