1
vote
0answers
13 views

Parking Lot Optimization Problem — How To Find the Minimal Path In A Periodic Set?

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 ...
0
votes
0answers
32 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
2
votes
1answer
74 views

How to calculate the point on the sphere that is nearest to some given points on the sphere?

Given some points $X=\{x_i:||x_i||=1,i=1,\ldots,n\} $ located on the sphere, how to calculate the point $\tilde{x}$ on the sphere that is nearest to these given points. That is to say ...
3
votes
1answer
136 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
1
vote
1answer
136 views

Minimum distance to points in plane

Someone told me that the the following problem is elementary. Given three points $a=(-5,0)$, $b=(0,5)$ and $c=(5,0)$ in $\mathbb R^2$ with Euclidean norm: $$\mbox{minimize}\;\; \; f(x)=\|x-a\| + ...
2
votes
1answer
29 views

Chord Maximisation

I am currently going back through all the "Challenge" questions in preparation for exams, and for this I do not know where or how to start, any hints would be appreciated. Now the way I have ...
2
votes
2answers
136 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
1
vote
0answers
21 views

Finding coordinates of nodes in a graph

I have a complete graph in which the edges represent the euclidean distance between the nodes which is known. Assuming a node to be (0,0), I want to find (approximately) the coordinates of other ...
0
votes
0answers
50 views

Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
17
votes
3answers
279 views

What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
0
votes
1answer
37 views

$|\langle a_i, a_j\rangle|$ for $p$ points on a unit circle.

Is it true that given any $p$ points $a_1, .., a_p$ on a unit [euclidean] circle, there is always a pair $i \ne j$ such that $|\langle a_i, a_j\rangle| \ge \cos{\pi/p}$?
0
votes
0answers
16 views

Dimension of face of a polytope in terms of linear functionals

A face of a polytope is a subset consisting of the points for which some linear functional is maximized. Can one determine the dimension of the face in terms of the linear functionals which are ...
0
votes
1answer
30 views

Highest Volume/Area Ratio

Given a fixed volume of a solid, what would be the shape of such solid that would minimize the its surface area? How to determine it? I thought about it, but I cannot find an algorithm that doesn't ...
2
votes
2answers
50 views

Find out minimize volume (V) of tetrahedral

I have this problem: On space $ (Oxyz)$ given point $M(1,2,3)$. Plane ($\alpha$) contain point $M$ and ($\alpha$) cross $Ox$ at $A(a,0,0)$; $Oy$ at $B(0,b,0)$; $C(0,0,c)$. Where a,b,c>0 Write the ...
1
vote
1answer
14 views

Placing a shape on a grid

I am interested in a certain kind of geometrical optimisation problems. I will illustrate it on a semi-concrete example: You are given a two-dimensional shape, say a polygon, and a rectangular ...
3
votes
2answers
68 views

The smallest quadrangle inscribed in a rectangle

I'm supposed to find a quadrangle of the smallest perimeter possible inscribed in a rectangle. The inscribed quadrangle has each of its four vertices on another side of the rectangle. Let's call the ...
2
votes
1answer
35 views

Shortest polygonal path inside a rectangle

I hope you could tell me if my reasoning is correct. We are given two points $A$ and $B$ inside a rectangle $PQRS$. We create a path $AXYB$ such that $X$ and $Y$ lie on different sides of this ...
1
vote
1answer
41 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
3
votes
1answer
50 views

Minimizing distance of circles from points without overlapping

I am designing a user interface, and I have encountered the following problem: I have $p_1 ... p_n$ points in $\mathbb{R}^2$, and $c_1 ... c_n$ circles with constant $r$ radius. I want to minimize ...
1
vote
3answers
46 views

Max perimeter of triangle inscribed in a circle

What is the maximum perimeter of a triangle inscibed in a circle of radius $1$? I can't seem to find a proper equation to calculate the derivative.
2
votes
1answer
44 views

How to minimally move circles so that they don't overlap?

You're given a set of circles, all the same radius, residing at different locations in a 2d space. Some circles are in fixed positions. How do you make sure none of them overlap, minimizing the ...
0
votes
1answer
30 views

Size of square formed by soap in a cube frame

So through the work of Plateau (as I understand it), we know that soap tries to find the shortest connection between points. At least, that's what I was taught. With this in mind, I had to solve the ...
0
votes
1answer
29 views

Maximizing the length of a right-triangle hypotenuse

Given different continuous ranges of values for the legs, how can I find the values that maximize the length pf the hypotenuse of the right triangle? In other words, given that A lies between X and Y ...
0
votes
0answers
33 views

How can I maximize the area of a rectangle given a continuous range of values for the length and width?

All the examples I have for rectangle area maximization problems start by having one of the sides fixed. But suppose I have a continuous range such that length is between A and B and height is between ...
1
vote
0answers
43 views

Maximum of the minimal distance of a set of points in an equilateral triangle

In this question, a closed triangle on a plane is a set of all points in its area and on its boundary, while an open triangle excludes its boundary. Now, the problems: Let $T$ be an equilateral ...
1
vote
2answers
45 views

is there a problem in the answer? finiding an angle

Can you tell me if there's an error in the answer? Given isosceles triangle $ABC$ ($AB=AC$) and $AB=b$. $BD$ is perpendicular to $AC$ and $DE$ is perpendicular to $BC$. Angle $BAC=2x$. The ...
0
votes
0answers
23 views

Maximization of the area of 2 irregular quadrilaterals formed at the intersection of 3 rectangles

Consider the intersections formed by a rectangle overlapping 2 other rectangles as shown in the blue outlined areas here; Given the following parameters; all rectangles are of an equal height ...
1
vote
1answer
33 views

$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My ...
2
votes
2answers
54 views

Maximal area in fixed perimeter [duplicate]

An old story I heard starts by two people that was arguing about how much land a man need. So they called to a young man, and said to him: You are stating in that point, and start running all the ...
2
votes
0answers
29 views

Extremal constant width curves on sphere

Definition Given some length $w\in\mathbb R$, I'm interested in closed convex sets $S$ of points with the following properties: For all pairs of points from $S$, the distance between them will be ...
2
votes
0answers
76 views

Megiddo's algorithm for lines of least weighted sum distance from a set of points

I came across the following problem: Given a set of n points (coordinate in 2d plane) within a rectangular space, find out a line ($ax+by=c$), from which the sum of the perpendicular distances of all ...
1
vote
3answers
87 views

Minimize the area of a triangle

Let $A \neq B$ be fixed points outside a fixed circle with centre $C$. The point $D$ can be chosen freely on the circle. The goal is to minimise the area of triangle $ABD$. Degenerate triangles ...
-1
votes
1answer
78 views

Lp optimal solution question

i have a general question. if there is a general LP problem $c^Tx$ s.t $A\cdot x \le b$, and $x \ge 0$ and assuming that the components of $c$ are non-zero entries then how can I prove that when $x$ ...
2
votes
1answer
110 views

Does the uniqueness of solutions to convex optimization with linear constraints hold in n>3 dimensions?

This is a repost of an earlier question, where I think I was not clear enough in what I was asking: I am examining the following optimization problem, for which I would like to know if, when a ...
5
votes
3answers
244 views

Minimizing the length of wire between two poles?

There are two poles (lets say poles A and B) $50$ feet apart and the poles are $15$ and $30$ feet tall. There is a wire which runs from the top of pole A to the ground, and then to the top of pole B ...
3
votes
0answers
39 views

Selecting k vectors with maximum spread out of a set of n vectors

Given a set $\mathcal{V}$ of $n$ vectors, find a subset $\mathcal{V}_k = \mathcal{V} - \mathcal{V}_{n-k}$ containing $k$ maximally spread vectors. Intuitively, these $k$ vectors should be spread as ...
2
votes
2answers
108 views

When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
38
votes
3answers
984 views

Where to build a bridge to cross a river in the shape of an annulus

There is a river in the shape of an annulus. Outside the annulus there is town "A" and inside there is town "B". One must build a bridge towards the center of the annulus such that the path from A ...
1
vote
1answer
184 views

How many n square can fit into a square of side N

Suppose we have n small squares of equal sizes that has area w. Suppose we have a fix square S of area A such that for area A, one area w < area A. If square S's area A, length, and width are ...
3
votes
1answer
147 views

Finding the extrema of $E(\vec{r})=\frac1a x^2+\frac1b y^2+ \frac1c z^2$ with respect to constraints geometrically

I have a function, $$E(\vec{r})=\frac1a x^2+\frac1b y^2+ \frac1c z^2.$$ Where $\vec{r}=(x,y,z)$ and $a>b>c>0$. I wish to find the maximum and minimum of this function with respect to the ...
3
votes
1answer
91 views

Largest scaled rotated rectangle inside rectangle

There are two rectangles: $r_1$ and $r_2$. $r_1$ is rotated $\theta$ and then uniformly scaled by a factor $k$ to exactly fit within $r_2$. I'm trying to find the value of $\theta$ that maximizes $k$, ...
0
votes
1answer
31 views

Making the Smallest Number of Mistakes Possible

I have the following problem. I have a set of $k$ labelled points, $\left\{\mathbf{x}_i, y_i\right\}_{i=1}^{k}$, where $\mathbf{x}_i\in \mathbb{R}^{2}$, and $y_i\in\left\{-1,1\right\}$. I want to ...
1
vote
2answers
81 views

Optimal Configuration for a Set of Points

Consider a set of $n$ points on the plane with positions $\mathbf{p}_1,\dots,\mathbf{p}_n$, such that each point $i$ has at least one neighbor $j$ at a distance of no more than $\lambda$ away from it ...
1
vote
1answer
143 views

Minimize the sum of distance under maximum norm

Given a set of points (Xi, Yi). I need to find a point (doesn't have to be in the given set) that minimize the sum of distance to the other points. The tricky part is the distance is measured by ...
0
votes
0answers
25 views

extremal points on a manifold intrinsiclly

I am wondering if there is a geometric object for real analytic manifolds that characterizes extremal points of the manifold intrinsically. For instance, suppose I live in the manifold, can I ...
2
votes
1answer
73 views

Optimizing Rectilinear Distance Traveled

I have a simple pipe network like this (not to scale): I can place a "valve" on any point on that pipe. What the valve does is it permits a certain viscous fluid to fill the pipes. However, because ...
18
votes
5answers
327 views

If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$?

Let each of $A, B, C, D, E$ be an angle that is less than $180^\circ$ and is greater than $0^\circ$. Note that each angle can be neither $0^\circ$ nor $180^\circ$. If $A+B+C+D+E = 540^\circ,$ what is ...
2
votes
1answer
142 views

Find a projection of a $k$-simplex with minimal “radius”

Let $S$ be a $k$-simplex in $\mathbb{R}^k$. I'd like to find a hyperplane $P$ that passes through the origin such that projections of vertices of $S$ onto $P$ are as close as possible to the origin ...
0
votes
0answers
40 views

Optimization and Duality: Maximizing surface / Minimizing Boundaries

I have recently gone trough an optimization course. I just thought about common problems I encountered in many domains. Given the measure of a boundary (lenght in 2D, surface in 3D), what shape ...
3
votes
0answers
86 views

largest empty sphere or rectangle

In N (~ 500) dimensions, I wish to find out the largest sphere or rectangle such that the sphere/rectangle does not contain already existing points. The entire set of points is bounded in an ...