# Tagged Questions

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### Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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### How to reduce 3 dimensional optimization to 2 dimensions?

I am trying to minimize the surface area of a parallelepiped of unit volume. Using Volume = xyz(1 + 2cos(a)cos(b)cos(c) - cos^2(a) - cos^2(b) - cos^2(c))^1/2 = 1 where x,y,z are edge lengths and ...
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### How to find the parallelepiped of unit volume with minimal surface area?

Is it best to approach this problem using edge lengths and the angles between them? I am trying to reduce the problem to two dimensions, although I haven't successfully done so yet So I have Volume ...
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### A point minimizing total great circle distance to a given set of points on a hemisphere

If you have a set of points on a hemisphere, how do you find a point on that hemisphere that has the minimum total great circle distance to the points in the set.
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### Largest Equilateral Triangle in a Polygon

Is there an algorithm to determine the largest equilateral triangle in a convex polygon?
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### Minimal volume of a tetrahedral

I'm unsure how to solve the following problem: Let $\textbf{p}=(a,b,c) \in \mathbb{R}^{3}$ with $a,b,c > 0$. For $\alpha , \beta > 0$ the equation $$\alpha (x-a)+ \beta(y-b) + (z-c) =0$$ ...
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### What is $s$ in s-energy (eg. Riesz s-energy)

I'm trying to understand fekete problems. There is a variable $s$ and a related concept of 's-energy' [1] [2] [3] [4] that comes up repeatedly when borrowing the concept of potential energy to find ...
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### Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
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### Convex hulls for a finite amount of points

I'm trying to understand what a convex hull intuitively is, and say given for a set of points $(x,y)\in\mathbb{R}^2$ how is it generated from these points? I tried reading the wikipedia article and ...
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### How fat is a triangle?

The slimness factor of a geometric shape in 2 dimensions is the ratio between the side-length of its smallest containing square and its largest contained square. This is an important factor in ...
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### Characterization of sphere.

I'm editing the question because I think the previous formulation was leaving a key element of the problem out and that was making it impossible to answer the question. I tried to update/improve the ...
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### Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
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### Solve for transform of rotating frame to fixed frame given points in rotating frame and a planar constraint

Say there are 2 coordinate systems, with one orbiting around the other. Call one fixed ƒ and the other rotating ρ. The goal is to find the transform between the two frames. What's known is A set of ...
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### How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
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### finding points with maximum distance between them on a circle

I'm a computer science student working on a problem in computer graphics and looking for a formula that can find the x and y positions of a set of N points on the surface of a circle so that the ...
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### Find the ellipse inscribed in a triangle having the maximum area

If we have a triangle of sides $a,b,c$, there are infinite ellipsis inscribed in the triangle. How can I find that having the maximum area? Is this ellipse the circle, or in what cases the maximum ...
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### Maximal area covered by two triangles in unit circle

What is the maximal area covered by two triangles in a unit circle? There are no restrictions other than that. They can overlap, touch the circle, not touch the circle etc. So far I have shown In ...
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### Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there any relationship between dual of an optimalization problem and dual of a polyhedra?
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### Maximizing $x\left(\displaystyle\sum_{k=0}^n \sqrt{1-kx^2}\right)$

Here's the motivation for my problem: Consider a right-angled triangle with legs of length 1 and $x$ and on the leg of length 1, and construct another right-angled triangle whose hypotenuse is 1 ...
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### 10 points inside a square - minimum distance between any of them

A square of side 1 is given, and 10 points are inside the square. If we divide the square into 9 smaller squares, and apply Dirichlet principle, we can prove that there are 2 of these 10 points whose ...
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### Platonic solids and charged particles

It is known that there are five Platonic solids: If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will ...
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### Prove That the Second Moment is Minimized with a Circle Packing

Graham and Sloane studied the problem of minimzing the second moment of disks on the plane, i.e. minimize $$U = \frac{1}{d^2} \sum_{i=1}^{n} || \mathbf{p}_i - \bar{\mathbf{p}} ||^2$$ s.t. ...
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### Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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### Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
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### How to find the minimal path between points in a planar set with holes in it?

When I was a commuter student, I would park in a very large parking that that had a set of stairs in a corner that I had to climb. In general, I had to park far away from this corner in an almost full ...