All Questions
18 questions with no upvoted or accepted answers
6
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219
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How big a box can you wrap with a given polygon?
Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of ...
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6
votes
0
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114
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Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces
Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
- 161
3
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0
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226
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Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap
I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...
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3
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65
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Cutting triangles into triangles with equal longest side
This post elaborates on a specific instance of Cutting convex polygons into triangles of same diameter .
Question: For any integer n, can any triangle be cut into n non-degenerate triangles all of ...
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3
votes
0
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260
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What is the VC-dimension of regular convex k-gons in the plane?
Recall the relevant definitions:
Let $H$ be a family of sets in $\mathbb{R}^d$. The intersection of $H$ with a point set $C$ is defined as $H\cap C = \{h\cap C\mid h\in H\}$. The VC-dimension of $H$ (...
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3
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141
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Optimal intersections between planar convex regions
Here is an earlier discussion that could be related:
On comparing planar convex regions of equal perimeter and area
We are broadly interested in placing two given planar convex regions so that the ...
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3
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0
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175
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Cutting convex polygons into triangles of same diameter
This question continues from: Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points ...
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3
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0
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169
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Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)
Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of $...
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2
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0
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154
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Reduced Voronoi diagram
I am currently reading Differentiable Surface Triangulation presented at Siggraph Asia 2021.
I think most of the paper is clear to me, though I keep re-reading through to see if I miss details.
The ...
- 115
2
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0
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117
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Folding polygons into 'vessels'
This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf
Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a ...
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2
votes
1
answer
504
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Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
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1
vote
0
answers
91
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A claim on the largest area circular segment that can be drawn inside a planar convex region
This post adds a little to To find the longest circular arc that can lie inside a given convex polygon
A circular segment is formed by a chord of a circle and the line segment connecting its endpoints....
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1
vote
0
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124
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A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
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1
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0
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46
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Multi-layered wrapping of polyhedra
This post continues from How big a box can you wrap with a given polygon? and Convex polyhedra that can be folded from convex polygons. One can also mention 'k-fold coverings of the plane' as examined ...
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1
vote
0
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48
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Deployment and dispersion in triangular regions
Definitions (from C. Stanley Ogilvy's 'Tomorrow's Math'):
Deployment: To place a specified number $n$ of points (stations) in a region such that the maximum distance of any point in the region from ...
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1
vote
0
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57
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Covering the annulus of symmetric convex body
Consider a symmetric convex body $A$ in $\mathbb{R}^d$. Now, we draw another object, $A'$, concentric and translated with respect to $A$ and having radius slightly greater than twice to the radius of ...
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0
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68
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Comparing partitions of a given planar convex region into pieces with equal diameter and pieces of equal width
We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
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0
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0
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93
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On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ...
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