2
votes
0answers
46 views

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and its smallest containing square has side-length 2. What is the smallest possible angle of the polygon? What is its smallest possible area? ...
6
votes
2answers
221 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
0
votes
1answer
34 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
57 views

Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
2
votes
0answers
132 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 ...
5
votes
1answer
100 views

Distance from point to vertices of convex hull

let $P = \{p_1, \ldots, p_k\}$ be any $k$ points on the unit $n$-sphere in $\mathbb{R}^{n+1}$ and $p_0 = 0$ the origin. Furthermore, let $CH(P\cup \{p_0\})$ denote the (possibly degenerate) convex ...
1
vote
1answer
16 views

Let $Ax = b$ be a system of hyper-planes that form a bounded convex $D$. Can $D$ be partitioned into union of adjacent simplices?

Let $Ax = b$ be linear system that forms $q$ bounded region $D$. If the columns of $A$ are independent, can $D$ be written as a union of adjacent simplices?
2
votes
2answers
144 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
91 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
141 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
49 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
233 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
104 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
232 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
556 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
144 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
110 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
260 views

How close are star-convex sets to convex sets?

What interesting properties of convex sets are retained by star-convex sets?