0
votes
0answers
21 views

Detecting faces of polytopes

I am working in convex geometry for the summer with little experience beforehand. It's a lot of fun but it does mean I don't know some of the basic things. I'm interested in the orbits of finite ...
0
votes
1answer
189 views

convex hull function in matlab

Is there anyway to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
0
votes
0answers
56 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, ...
0
votes
1answer
19 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
40 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
31 views

Inequality description of Convex hull

Given a finite collection of points $p_1,p_2,\ldots,p_m \in \mathbb{R}^n$, what are the inequalities describing their convex hull $\text{Conv}\{p_1,p_2,\ldots,p_m\}$?
2
votes
1answer
74 views

Does the center of a convex region lie within that region?

There's probably a simple result that says this is true, but I sure can't find it. It seems obvious, though. Let $D$ be a closed, compact region in $\Re^n$. Further, let $D \subseteq [0,l]^n$ and ...
0
votes
1answer
34 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
45 views

Explain about convexity in geometry and in optimization.

My question is 'what is a difference between convexity in geometry and optimization?'
1
vote
0answers
127 views

Facets of the convex hull as solution of an optimization problem?

Given $N$ points $x_1, x_2, ..., x_N \in \mathbb{R}^n$, consider their convex hull $$\mathcal{C} = \text{conv}( \{ x_1, ..., x_n \} ) = \bigcap_{j=1}^{J} \{ x \in \mathbb{R}^n : \ A_j x \leq b_j \} ...
3
votes
2answers
187 views

Computing the decay factor for a full rank wide matrix, or finding a unit vector farthest away from a set of spanning unit vectors

Let $A$ be a tall matrix that is not rank-deficient and has normalized columns. That is $A$ is $m\times n$, $m<n$ and rank$(A)=m$, and $||a_i||_2=1$ for all columns $a_i$. Define ...
0
votes
1answer
29 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 ...
0
votes
2answers
189 views

distance between a polytope point and a polytope vertex

How to find distance in between any polytope point to the closest vertex of the polytope (the verteces of the polytope are known)? How to find a distance from the farest polytope point to the closest ...
5
votes
1answer
124 views

a conjecture on norms and convex functions over polytopes

Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
4
votes
1answer
208 views

A conjecture of parallelogram inside convex and central symmetric curve

Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$. How to prove the conjecture that $\displaystyle ...