Tagged Questions
2
votes
2answers
93 views
Sets whose intersection with line segments have finite components
Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
3
votes
1answer
55 views
A particular ILP where the existence of a relaxed solution implies the existence of an integer solution
This question emerged from a discussion on my previous question Determining quickly whether this Integer Linear Program has any solution at all, which I would like to elaborate separately.
I am ...
1
vote
1answer
91 views
Separation Theorem in Euclidean Space.
I want to show the following:
Let $A,B \subseteq \mathbb{R}^n$ disjoint, nonempty, closed and convex sets. Then there exists a $h \in \mathbb{R}^n$, such that $A$ and $B$ gets separated in the ...
1
vote
1answer
46 views
Does the following condition characterise convexity of a set?
Conjecture: A set $X \subseteq \mathbb{R}^n$ is convex if and only if the following holds.
For any $x \in X$ and any vector $v \in \mathbb{R}^n$ such that $x+v \notin X$, it holds that for any scalar ...
0
votes
1answer
121 views
Intersection of Two Simplices
How to find vertices a the polytope-intersection of two simplices, if I know the vertices of these simplices.
More precisely:
Let $T_1$ and $T_2$ be two regular $n-1$ dimensional simplices with ...
2
votes
1answer
82 views
Intersection of 2 $p$-simplices is a finite union of some $p$-simplices
I'm looking for a non-painful proof of this assertion.
A p-simplex is defined as the set of all sums $\sum_{i=0}^p t_i x_i$ with $0\leq t_i\leq 1$, $\sum_{i=0}^p t_i=1$ for a geometrically ...
5
votes
2answers
95 views
condition for a set to be compact and convex
Is it true that a set in, say, $n$-dimensional Euclidean space is compact and convex iff its intersection with any line is empty, a single point, or a closed line segment?
10
votes
2answers
390 views
If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?
If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
4
votes
3answers
126 views
Is concavity of a real-valued function on a Euclidean space implied by concavity of its restriction to every lower dimensional affine subspace?
Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over ...
1
vote
2answers
89 views
Radius of a hypercube at a given angle
For a ray from the origin with a given angle in $R^n$, I am trying to find the radius at which that ray intersects the frontier of the unit n-cube. In two dimensions, the picture is this:
Given ...
1
vote
1answer
215 views
How close are star-convex sets to convex sets?
What interesting properties of convex sets are retained by star-convex sets?
