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angle constraint on convex hull

Given a set of points $P$ in $R^d$ it is straightforward to compute the convex hull (Graham-scan etc). However, the angle between the adjacent faces are unconstrained (apart from necessitating that ...
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volume of convex hull given boundary

Suppose we are given a boundary $ \Gamma \in R^3$ (the boundary not necessarily being convex) and asked to estimate the volume of the convex hull, can we do so without actually computing the convex ...
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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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1answer
69 views

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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1answer
48 views

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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1answer
44 views

Lower bound for point set triangulation

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 $\...
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1answer
76 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 ...
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1answer
57 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 ...
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420 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 ...
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62 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 ...
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1answer
355 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 ...
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1answer
39 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 ...
2
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1answer
85 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,...
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341 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 ...
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1answer
495 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 ...
5
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1answer
104 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 ...
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86 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 ...
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1answer
93 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
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1answer
182 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 ...
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0answers
35 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 ...
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1answer
64 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, ...
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1answer
85 views

How to use convex hull for this problem

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 ...
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1answer
347 views

Understanding a few intricacies related to two naive algorithms to compute the convex hull of a set of points

I already understood how the well-known algorithms like Graham's scan, Quickhull, etc. work, but I have difficulties in understanding 2 naive algorithms to compute the convex hull: Let $S= \{p_1, \...