# Questions tagged [convex-hull]

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### Difference between convex hull algorithms

I was wondering what are the main differences in terms of efficiency of convex hull algorithms? Brute force algorithm is inefficient due to iterating over every three vertices in our set and its time ...
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### Upper and lower tangent line to convex hull from a point

Is it possible to find an upper and lower tangent line to a convex hull in $log(n)$ time where $n$ is number of points on a convex hull? I have just done it in linear time where I checked for upper ...
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### Checking if a given point is convex hull vertex [duplicate]

I have a problem in computational geometry: Given $n$ point in 2D space,and given a point $P$, design an algorithm check that whether $P$ is a vertex of convex hull or not in $O(n)$. My idea 1: I ...
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### Maximize length of side of triangle from points on a circle

Given a circle with $n$ points, among all triangles we can make using these points, we want to find a triangle with maximum length of its shortest side in $o(n^2)$. We try to make a relation between ...
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### Efficient algorithm to compute the diameter of a convex set?

Is there a polynomial algorithm that can compute the diameter (the distance between the furthest points) of a convex set? It is possible to do it efficiently for a set of points, but imagine that the ...
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### minimum number of points a convex hull must have

Quick question: Say for example there are 10 colinear points. my question is does a convex hull have to be a convex polygon? or can it be a line as well according to the formal definition of the ...
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### Integer Linear (ILP / MILP) Formulation for Collision Avoidance of Convex Polytopes / Polyhedra

I am looking for a possibility to avoid the collision of two convex polytopes using (mixed integer) linear programming. I know how I can detect a collision (Akgunduz, A., Banerjee, P., and Mehrotra, S....
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### minimum moves to be made to get out of convex hull

Given a convex hull in $XY$ plane. and we have $n$ points sitting inside it. In one move we can move all the points to right by $1$ or move all the points to up by $1$. what are the minimum number of ...
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### Concentric convex hulls

Given N points in a 2D plane, if we start at a given point and start including points in a set ordered by their distance from the starting point. After including every point, we check if there is a ...
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### Finidng edges of convexhull from rectangles

Green Boxes = rectangles red dots = edge points red lines = to be generated from convex hull algorithm I have a problem with creating the convex hull algorithm. I want to select or collect all the ...
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### Minimal set of inequalities including good points but excluding bad points

Suppose I have a collection of good convex sets and bad convex sets in $\mathbb{R}^d$ (where $d$ can be big). Each convex set is defined by a series of closed ranges in each dimension $d$ - a ...
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### Distance from high dimensional convex hull to target point T

I have a set S of high dimensional points in Euclidean space, with convex hull H (not known); and some target point T in that space not in or on H. Rather than worry about calculating both H and the ...
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### Convex hull in a discrete space

I know some algorithms which compute the convex hull in a continuous space. Are there efficient algorithms to compute it in a discrete domain? For example in 3D discrete space, given the blue points, ...
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### If a convex optimization problem can be NP-Hard, in what sense are convex problems easier than non-convex problems?

Being new to the OR and Optimization world, I've always assumed that a problem being convex meant that it can be solved in polynomial time. Now I am learning that a convex optimization problem can ...
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### 3D gift wrapping algorithm: how to find the first face in the convex hull?

I am implementing the gift wrapping algorithm to find the convex hull of a set of points in the 3D space. However, all the articles I have read seem to omit the description of the first step of the ...
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### Is binary-search really required in Chan's convex hull algorithm?

I have a little doubt about Chan's algorithm. From Wikipedia's description we see that the second phase of an algorithm works with $K = \mathrm{ceil}(\frac{n}{m})$ subsets $Q_i$. The goal of the ...
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### Minimum distance between two convex hulls maximized

I want to implement a program that splits a set $S$ of $n$ points in the plane into two sets such that the distance of the convex hulls of the two sets is maximized. It should be done in $O(n^3)$. I ...
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### Why is the graph inside Graham Scan always planar

One of the ways to prove that Graham Scan constructs convex hull in linear time is using planarity of the graph obtained by running the algorithm. This graph is always planar, so according to Euler's ...
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### Convex Hull for known output size in O(n)

Given a set $S$ of $n$ planar points, we know that the $|CH(S)| = 17$. How can I create an algorithm that computes $CH(S)$ in $O(n)$ time? Why is it really $O(n)$?
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How can I show that the lower bound for getting a triangulation from a point set in the plane is $\Omega(n \log n)$? I know that the lower bound for finding the convex hull for a point set is also $\... 1answer 217 views ### How to prove that non-antipodal vertices cannot be a diameter of a convex polygon? I am learning Shamos's rotating calipers algorithm for finding the diameter of a convex polygon in his Ph.D. thesis; Page 78. It reads Consult Figure 3.23 and notice that parallel lines of support ... 1answer 156 views ### Convex hull of fixed size I have heard that the quickhull algorithm can be modified if the size of the convex hull (the number of points it consists of) is known beforehand, in which case it will run in linear time. What ... 1answer 520 views ### How to find the supremum over all the “good” (interior) polytopes for a given set of 3D points? Let$S \subset \mathbf{R}^3$be a set of points in 3D and let$O=(x_0,y_0,z_0)$be the origin/point of reference. We consider a convex polytope$P$good / interior if:$P$is wholly contained ... 0answers 185 views ### Nested Convex Hulls Algorithm The convex hull of a point set is a well understood problem and nice optimal solutions are known in the case of a finite point set and a simple polygon. For a convex polygon, the hull is the polygon ... 1answer 851 views ### The use of binary search when determining whether a point lies inside a given convex hull In an answer to the problem of determining whether or not a point lies inside a given convex hull, a thesis is mentioned, which says : For repeated queries with preprocessing allowed, we develop a ... 1answer 79 views ### How to convert two conflicting objective functions into a single objective function I have two objective functions say f1 where I have to minimize (X+Y) and another function f2 where I have to maximize (A-B). The two functions are conflicting. I need to convert them into a ... 1answer 144 views ### Internal tangent intersection of two point sets in linear time I need to find the intersection of the internal tangents of two point sets$V_a, V_b$in$\mathbb{R}^2$, defined via their convex hulls. We can assume that the sets are disjoint and linearly separable,... 0answers 697 views ### Finding the Convex Layers of a given set of points Definition of convex layers can be found at wikipedia. I was trying to understand this algorithm , which works in O(n log n) time, which is optimal. In the paper, the author has described two ... 1answer 1k views ### Convex-hull of a star shaped polygon in O(n) I'm trying to develop an algorithm to find a convex hull of a star shaped polygon in$O(n)$time. The specific problem, which is also described here, is as follows: A polygon$P$is star-shaped if ... 1answer 183 views ### Optimization over convex combinations in a circle Consider the following situation: given a triangle$ABC$inscribed in a circle, define$f$as the product $$f(P) = d(P, A) \; d(P,B) \; d(P,C)$$ where$P$is a point on the circle and$d$are ... 0answers 98 views ###$3$-dimensional convex hull using only a desired number of planes I would like to find the convex polytope with the smallest volume that envelops (contains) all the points of a given 3D point cloud and that can be constructed from only$k$planes. This is similar to ... 1answer 159 views ### Convex Hull Problem proof [closed] In a given set of points, Prove that the two farthest points are the vertices of the convex hull, How can i get the accurate proof, so that the question can be explained in the class 1answer 369 views ### Convex hull algorithm in$O(\min(mn, n\log n))$I am looking for an algorithm to compute the convex hull of a set of$n$points$P$. The hull should contains$m$points. This algorithm should work in time$O(\min(mn,n \log n))$. My first guess was ... 0answers 45 views ### Does Optimal Substructure implies Convexity and vice versa? In undergraduate CS, Dynamic Programming problems are often related to Overlapping Optimal Substructure (https://en.wikipedia.org/wiki/Optimal_substructure). Dynamic Programming is also often used in ... 1answer 224 views ### About the Max-Cut SDP The Max-Cut optimization problem on a graph$G=(V,E)$can be written as the question of wanting to maximize the function$\frac{1}{4} \sum_{(i,j) \in E } (x_i -x_j)^2$under the constraint$x_i^2 = 1, ...
Problem: You are to collect a total of $N$ litres of red and $M$ litres of blue liquid. For doing job $i$ for time $t$, you get $a_i t$ litres of red and $b_i t$ litres of blue liquid. $t$ need not ...