Related papers: Tightly Circumscribed Regular Polygons

Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles…

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a…

We generalize regular subdivisions (polyhedral complexes resulting from the projection of the lower faces of a polyhedron) introducing the class of recursively-regular subdivisions. Informally speaking, a recursively-regular subdivision is…

In this paper, we propose a topology optimization (TO) framework where the design is parameterized by a set of convex polygons. Extending feature mapping methods in TO, the representation allows for direct extraction of the geometry. In…

Let $P$ be a convex polygon in a Hadamard surface $M$ with curvature $K$ satisfying $-k_2^2 \ge K \ge -k_1^2$. We give an upper bound of the circumradius of $P$ in terms of a lower bound of the curvature of $P$ at its vertices.

Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian…

This paper proposed a method to judge whether the point is inside or outside of the simple convex polygon by the intersection of the vertical line. It determined the point to an area enclosed by two straight lines, then convert the problem…

We propose a technique called Rotate-and-Kill for solving the polygon inclusion and circumscribing problems. By applying this technique, we obtain $O(n)$ time algorithms for computing (1) the maximum area triangle in a given $n$-sided…

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically…

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence…

The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control…

This second part on polygons in the hyperbolic plane is based on the first part which deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The topic here is the maximum question for the area of these…

We give a sharp lower bound to the largest possible Euclidean norm of signed sums of $n$ vectors in the plane. This is achieved by connecting the signed vector sum problem to the isoperimetric problem for the circumradius of polygons. In…

Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have…

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…

We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal…

Given a manifold $\mathbb{M}$ admitting a maximally supersymmetric consistent truncation, we show how to formulate new consistent truncations by restricting to a set of Kaluza-Klein modes on $\mathbb{M}$ invariant under some subgroup of the…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new…

A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive…