1
vote
1answer
39 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
1
vote
3answers
63 views

Is convex hull of a finite set of points in $\mathbb R^2$ closed?

Is the convex hull of a finite set of points in $\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!
2
votes
1answer
44 views

Relation between mean width and diameter

Question: Let $A$ be a compact set in $\mathbb R^n$. Is it always true that $\text{mean-width}(A) \ge C \cdot \text{diam}(A)$ for some constant $C$ depending only on the dimension? If not, is it ...
0
votes
1answer
17 views

How do I justify that a second order cone is an intersection of half space

I am studying convex optimization right now, and the text book claims that a second order cone is a collection of intersections of half space $$K_n = \bigcap_{ u:\|u\|_2 \leq 1} \{(x,y) \in R^{n+1 } ...
1
vote
1answer
29 views

Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
0
votes
1answer
35 views

Volume of unit n-dimensional ball, definite integal

As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true: ...
2
votes
1answer
30 views

Distance of convex combination of pairs of points in $\mathbb{R}^n$

Given 4 points $w,x,y,z \in \mathbb{R}^n$ define for $t\in [0,1]$ $f(t)=d(wt + (1-t)x, yt + (1-t)z)$. Is this function convex? I have found a proof by differentiating twice and calculating a lot but ...
0
votes
0answers
19 views

Folding in $\mathbb{R}^2$ - which area of modern geometry formalises this concept?

For any lines $L_1,...,L_n$ in $\mathbb{R}^2$ and $w_1,...,w_n \in \{-1,+1\}$ we can define a folding operation $F(L_1,...,L_n; w_1,...,w_n): \mathscr{P}(\mathbb{R}^2) \rightarrow \mathbb{R}^2$, ...
3
votes
1answer
61 views

Algebraic Proof that a Disk is Convex

After searching on Google for a while, I cannot seem to find an algebraic proof that a disk is a convex set. Intuitively, this seems obvious: if you take any two points $x, y$ in a disk, then the line ...
1
vote
1answer
40 views

Sets defined by distance to a convex set

Let $Y \subset \mathbb R^n$ be a bounded convex set, let $R>0$, and let $$Z := \left\{z \in \mathbb R^n : d(z,Y) > \dfrac12R \right\}$$ where $$ d(z,Y) = \inf_{y\in Y}|y-z|. $$ If you like, ...
0
votes
0answers
16 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
4
votes
3answers
189 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
2
votes
2answers
21 views

Convex set for a set of points in 2d plane

There are set of five points $A(0,0) ,B(1,1) ,C(2,0) ,D(2,2).E(0,2) F(1.5,1.5)$ $S=\{A,B,C,D,E,F\}$ Please tell me whether my understanding is correct or not! The points $A,C,D,E$ forms a convex ...
0
votes
0answers
27 views

How to find a curve inside a non-convex

I want to connect two points in a space within the space. If the space is convex, I can simple draw a line between them. But how about a non-convex space. How can I find a curve connecting these two ...
2
votes
0answers
93 views

Voronoi-Diagram - split concave polygon

I have a concave polygon c and a set of points p[]. What I need is to have a set of polygons splitted by the voronoi-diagram. My problem is the following: I already found a library to calculate the ...
0
votes
1answer
33 views

Geometry question with convexity

Assume that a function $h(\lambda)$ is decreasing and convex given interval $[l,u]$ and has an unique root $\lambda^*\in (l,u)$. Also, assume $|l-\lambda^*| > |\lambda^*-u|$. Consider any $z\in ...
4
votes
0answers
59 views

Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. ...
1
vote
1answer
55 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
1
vote
0answers
22 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
10
votes
1answer
67 views

Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what ...
1
vote
0answers
68 views

Relative Interior and dense subsets

(Due to no answers, I also posted this question here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ ...
1
vote
1answer
19 views

Formula for extracting bounding points from a given set of points

I have a set of geographic locations, i.e. points defined by latitude and longitude. Given this set of points, I need to select only those of them that belong to the surface bounds. The simplest ...
1
vote
1answer
41 views

Does a discrete set of points in $\mathbb{R}^{n}$ define a locally finite collection of hyperplanes?

Let $v_{1},v_{2},...$ be a discrete set of non-zero vectors in $\mathbb{R}^{n}$. By discrete, I mean that any $v_{i}$ is surrounded by an $\epsilon$-ball not containing any other point $v_{j}$. ...
2
votes
0answers
129 views

Infimum over area of certain convex polygons

Let us identify the Euclidean plane with the complex plane $\Bbb C$. For every $n\ge 1$, consider the following collection of finite sets in $\Bbb C^{n+1}$: $$S_n:=\{(z_0,z_1,\dots,z_n)\in\Bbb ...
3
votes
0answers
51 views

Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
1
vote
0answers
36 views

Convex polyhedron is union of simplices

Given a convex polyhedron $P$, how can we prove that every point $x \in P$ is in some simplex whose vertices are vertices of $P$? One proof is to inductively build a triangulation of $P$. If $P$ is ...
5
votes
1answer
178 views

Why does the amoeba shrink to its skeleton when we go to infinity?

Let $f\in\mathbb{C}[X_1^{\pm1},\ldots,X_n^{\pm1}]$ a Laurent polynomial. Let $\mathrm{Log}:(\mathbb{C}\setminus\{0\})^n\to\mathbb{R}^n$ defined by ...
1
vote
2answers
66 views

Number of faces of convex hull

If you have $n$ points in $d$-dimensional Euclidean space, the number of faces of the convex hull is potentially exponential I understand. How can this be proved?
6
votes
0answers
77 views

n-simplex in an intersection of n balls

Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
1
vote
1answer
84 views

A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
3
votes
1answer
158 views

Find the maximum convex area

My question is very similar to Plow's Question; but with this difference: How can I find the maximum convex area that can fit inside a non-convex region? For an example, consider this non-convex ...
4
votes
2answers
163 views

Volume of the projection of the unit cube on a hyperplane

Let $C_n\subset\mathbb{R}^n$ be the $n$-dimensional cube with side $1$, and let $P_k$ be any $k$-dimensional plane, $k\leq n$. What is the maximal $k$-volume $V_{n,k}$ of the projection of $C_n$ on ...
2
votes
1answer
131 views

Support function of a convex domain

Let $Q$ be a compact, convex domain in the plane, with smooth boundary $\partial Q$. We further assume that the origin is contained in $Q$. For a concrete example, let's take an ellipse ...
5
votes
0answers
134 views

What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? ...
1
vote
1answer
53 views

Convex set, differentiable in one point, inner direction

I have a convex set $\mathcal{C}$ with non-empty interior and I have shown that its boundary is differentiable in a point $c^*$, in the sense that there is a vector $n^*$(the normal vector), such that ...
3
votes
2answers
60 views

Need a way of computing the vertices of intersection of two simplices

I have two simplices $\Delta_1, \Delta_2$ defined as: The first simplex, $\Delta_1$, is the set of points defined as follows: $$\Delta_1 = \left\{\sum\theta_iu_i, \theta_i >= 0, ...
12
votes
2answers
479 views

Two cubes in unit cube

A cube of side one contains two cubes of sides $a$ and $b$ having non-overlapping interiors. How to prove the inequality $a+b \le 1?$ The same question in higher dimensions.
5
votes
1answer
171 views

Characterisation of linearly separable points of a hypercube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. E.g. for a cube, the following 4 points (red) are not linearly ...
1
vote
1answer
42 views

Explain about convexity in geometry and in optimization.

My question is 'what is a difference between convexity in geometry and optimization?'
2
votes
2answers
89 views

Given some points in the Euclidean space, find a plane satisfying some restrictions

In a 3-D Cartesian coordinates, suppose we are given $n$ points and their coordinate values in the form $(x,y,z)$. Obviously there are uncountable planes which divide those points into two groups, ...
0
votes
1answer
29 views

Find a point in a polytope that always cuts off a constant fraction of the polytope.

I have $k$ linear inequalities in $\mathbb{R}^n$, which we can express as $Ax \ge c$ (where $A \in \mathbb{R}^{k \times n}$ and $c \in \mathbb{R}^k$). Assume that the set of $x \in \mathbb{R}^n$ that ...
1
vote
1answer
48 views

Convex hull area of projected points are convex respect to rotations?

Let $A$ be a finite list of points in $\mathbb{R}^3$ and $c$ the centroid of $A$. Let $P$ be an orthographic projection onto a plane in $\mathbb{R}^2$ and $h$ be the convex hull of $P(A)$. Let further ...
2
votes
1answer
174 views

relative interior, convex hull, intersection

For any index set $I$, let $A_\iota\subseteq\mathbb{R}^d$ for $\iota\in I$ be closed sets. Do we have $\bigcap_{\iota\in ...
1
vote
1answer
79 views

affine set definition equivalence

How to show the following definitions are identical for an affine space: $C = p + W$ where $W$ is a subspace $p$ is a vector in $\mathbb{R}^n$, and $\lambda a + (1-\lambda) b$ is in $C$ for any $a$ ...
1
vote
0answers
116 views

How to generalise a result regarding intersections of cones and other convex sets?

To test for a particular property of positive LTI systems using feasibility problems I've come across the following claim which, intuitively, I believe can be generalised. I think I've (rather ...
3
votes
1answer
120 views

Intersection of two $n$-dimensional quadratic inequalities?

I have two quadratic inequalities of the form $$ a_1x^TAx + b_1^Tx + c_1 \le 0\\ a_2x^TAx + b_2^Tx + c_2 \le 0 $$ where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, ...
3
votes
1answer
108 views

Edges of the convex hull of a finite point set

I'm looking for a formal proof of a statement that seems obvious (and yet may be wrong) to make sure one of my algorithms is correct. Given a set S of N points in $\mathbb{R}^3$, suppose we have a ...
0
votes
1answer
27 views

Intersection of a 2-Dimensional body and a Line given west-most point and south-most point

I have a 2-Dimensional Closed Convex Compact Body $\mathbb{S}$(a set, for eg, a circular disc)). Assume, it has a non-zero intersection with the all-negative quadrant. Consider the following 2-D point ...
4
votes
1answer
459 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
10
votes
1answer
258 views

What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...