Related papers: Applications of Convex Analysis to the Smallest In…
In this work we study the fencing problem consisting of finnding a trisection of a 3-rotationally symmetric planar convex body which minimizes the maximum relative diameter. We prove that an optimal solution is given by the so-called…
Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming…
In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal. This problem was solved by…
A triangulation of a circle bundle $ E \xrightarrow[\text{}]{\pi} B$ is a triangulation of the total space $E$ and the base $B$ such that the projection $\pi$ is a simplicial map. In the paper we address the following questions: Which…
We study the problem of minimum enclosing rectangle with outliers, which asks to find, for a given set of $n$ planar points, a rectangle with minimum area that encloses at least $(n-t)$ points. The uncovered points are regarded as outliers.…
We consider geodesically convex optimization problems involving distances to a finite set of points $A$ in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the…
The problem of packing equal circles in a circle is a classic and famous packing problem, which is well-studied in academia and has a variety of applications in industry. This problem is computationally challenging, and researchers mainly…
In this paper, we consider the problem of computing the smallest enclosing ball (SEB) of a set of $m$ balls in $\mathbb{R}^n,$ where the product $mn$ is large. We first approximate the non-differentiable SEB problem by its log-exponential…
We investigate the problem of finding the minimum number of pieces necessary for dividing a three-dimensional sphere or a ball and reassembling it to form $n$ congruent copies of the original object, generalising a known result by Raphael…
The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject…
In the MINIMUM CONVEX COVER (MCC) problem, we are given a simple polygon $\mathcal P$ and an integer $k$, and the question is if there exist $k$ convex polygons whose union is $\mathcal P$. It is known that MCC is $\mathsf{NP}$-hard…
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle…
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…
Given a convex region in the plane, and a sweep-line as a tool, what is best way to reduce the region to a single point by a sequence of sweeps? The problem of sweeping points by orthogonal sweeps was first studied in [2]. Here we consider…
The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length…
We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak in the case of quadratic map. We…
Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
It has recently been established byWang and Xia [WX] that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in…
In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin…