Related papers: Envelopes of $\alpha$-sections

A polycube is an orthogonal polyhedron composed of unit cubes glued together along entire faces, and homeomorphic to a sphere. A layer of a polycube refers to the portion lying between two horizontal cross-sections spaced one unit apart. We…

The idealized theory of quantum vacuum energy density is a beautiful application of the spectral theory of differential operators with boundary conditions, but its conclusions are physically unacceptable. A more plausible model of a…

In this work, we construct some irreducible components of the space of two-dimensional holomorphic foliations on $\mathbb{P}^n$ associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. We give a…

Classic cake-cutting algorithms enable people with different preferences to divide among them a heterogeneous resource (``cake''), such that the resulting division is fair according to each agent's individual preferences. However, these…

Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…

What is the shape of the 2D convex region P from which, when 2 mutually congruent convex pieces with maximum possible area are cut out, the highest fraction of the area of P is left over? When P is restricted to the set of all possible…

Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the…

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems…

We study the computational complexity of fair division of indivisible items in an enriched model: there is an underlying graph on the set of items. And we have to allocate the items (i.e., the vertices of the graph) to a set of agents in…

Let $X$ be a set, $\ka$ be a cardinal number and let $\iH$ be a family of subsets of $X$ which covers each $x\in X$ at least $\ka$ times. What assumptions can ensure that $\iH$ can be decomposed into $\kappa$ many disjoint subcovers? We…

This is an experimental case study in real algebraic geometry, aimed at computing the image of a semialgebraic subset of 3-space under a polynomial map into the plane. For general instances, the boundary of the image is given by two highly…

Let P be a planar n-point set. A k-partition of P is a subdivision of P into n/k parts of roughly equal size and a sequence of triangles such that each part is contained in a triangle. A line is k-shallow if it has at most k points of P…

We initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each…

A certain real number, depending on two neighbouring sides of a quadrilateral and the diagonal meeting these two sides at their common point, is shown to be invariant under affinity. As an application we demonstrate a nice formula for the…

We prove that strictly convex surfaces moving by $K^{\alpha/2}$ become spherical as they contract to points, provided $\alpha$ lies in the range $[1,2]$. In the process we provide a natural candidate for a curvature pinching quantity for…

We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…

This article deals with the cake cutting problem. In this setting, there exists two notions of fair division: proportional division (when there are n players, each player thinks to get at least 1/n of the cake) and envy-free division (each…

In this paper, we study a reverse isoperimetric inequality for planar convex bodies whose radius of curvature is between two positive numbers 0 < $\alpha$ < $\beta$, called ($\alpha$, $\beta$)--convex bodies. We show that among planar…

Recently, Mr\v{s}evi\'{c} and Reilly discussed some covering properties of a topological space and its associated $\alpha$-topology in both topological and bitopological ways. The main aim of this paper is to investigate some common and…

We define a class of $L$-convex-concave subsets of $\mathbb{R}P^3$, where $L$ is a projective line in $\mathbb{R}P^3$. These are sets whose sections by any plane containing $L$ are convex and concavely depend on this plane. We prove a…