A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...
Questions tagged [convex-geometry]
1007 questions
41
votes
6 answers
Approximating a convex disk by an ellipse
For every convex compact set $K$ of area $1$ in $\mathbb{R}^2$, among all ellipses of area $1$ there exists an ellipse $E$ such that the area of the symmetric difference between $K$ and $E$ is smallest possible.
Questions.
(a) Is $E$ unique?
(b) If…
Wlodek Kuperberg
- 7,256
39
votes
2 answers
Abstract definition of convex set
I'd like to formulate an abstract definition of convex sets: a set $K$ is convex if it is endowed with a ternary operation $K\times[0,1]\times K\to K$, written $(x:t:y)$, satisfying axioms
$(x:0:y)=(x:t:x)=x$
$(x:t:y)=(y:1-t:x)$
$(x:t:(y:\frac…
grok
- 2,489
28
votes
2 answers
Can we always shift two disjoint convex bodies a little bit to decrease the volume of their convex hull?
Let $K,L\subset\mathbb R^d$ be two disjoint compact convex sets with non-empty interiors. Can $x=0$ be a point of local minimum for the function $F(x)=\text{vol}_d(\text{conv(K,L+x))}$?
I was asked this question by Dan Florentin a few weeks ago and…
fedja
- 59,730
10
votes
4 answers
When is the convex hull of two space curves the union of lines?
I am interested in the convex hull of two space curves. Let $A,B\subset \mathbb R^3$ be two spaces curves. I am interested in when $\mathrm {con} (A \cup B)$ equals to $$\bigcup _{a\in A,b\in B} \{\lambda a + (1-\lambda) b | 0\le \lambda \le 1\}.$$…
Gheehyun Nahm
- 909
9
votes
2 answers
Ellipsoid minimizing Banach-Mazur distance to convex body
Given a (symmetric) convex body $K \subset \mathbb{R}^n$ (equivalently, given a norm on $\mathbb{R}^n$), there is a unique ellipsoid of maximal volume in $K$, called the John ellipsoid. The John ellipsoid can be described as a ``canonical…
mdr
- 527
8
votes
2 answers
Is Minkowski sum of boundary convex again?
Consider a closed, bounded and convex set $C \subset \mathbb{R}^{2}$ and denote its boundary with $\partial C$. It is very well-known that the Minkowski sum of two convex sets is convex again. What about the Minkowski sum of its boundary?
Is the…
Hermann
- 91
8
votes
2 answers
An affine characterization of ellipsoids?
Let $K$ be a convex body of volume 1 in $\mathbb{R}^n$ and $x$ a (variable) point on the boundary of $K$. Define $f_K(x)$ to be the volume of the convex hull of the union of $K$ with its reflection in $x$, that…
Wlodek Kuperberg
- 7,256
8
votes
3 answers
Helly's theorem in other areas of mathematics
Are there some outstanding results using some version of Helly's theorem in a totally different area (whatever that means) than convex geometry?
7
votes
2 answers
Isodiametric hull
Let A be a convex compact set in the plane (with a piecewise smooth boundary, say). We want to `inflate' it in such a way that the diameter does not increase.
More accurately, we are looking for all sets C such that
a) A is a subset of C;
b)…
Nikita Sidorov
- 2,085
7
votes
2 answers
Applications of Cauchy's Arm Lemma
Cauchy's Arm Lemma is used in the proof of Cauchy's Rigidity Theorem for convex polyhedra. The Lemma states that in the plane or on the sphere that if all but one of the side lengths of two convex polygons $P$ and $P'$ are the same, and the angles…
Gordon Williams
- 153
6
votes
1 answer
If a compact convex set meets the positive orthant does it meet it at a point with a normal in the positive orthant?
Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R_+^n$. Does $C\cap \mathbb R_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R_+^n$ is normal to $C$ at $x$?
Here, a vector…
Alexander Pruss
- 2,383
6
votes
0 answers
A sufficient condition for being the boundary of one's convex hull?
Let $A\subset\mathbb R^n$ be such that:
every non-zero linear functional is maximized by a unique point of $A$
every point of $A$ is a point where some linear functional achieves its maximum over $A$ (i.e., every point of $A$ is an exposed point,…
Alexander Pruss
- 2,383
6
votes
1 answer
Average Volume of Convex Hull of N points in Unit Hypercube
Suppose we randomly pick $N$ points inside the unit hypercube in $\mathbb{R}^{N}$. What is the expected value of the volume of the convex hull (in any $\mathbb{R}^{N-1}$ containing the convex hull)?
For example, if $N=2$, we are asking about the…
user38495
- 1,052
5
votes
1 answer
Section of ellipsoids
I am reading some survey about euclidean sections of convex bodies(http://arxiv.org/abs/1110.6401). It is written that
any $k$-dimensional ellipsoid easily seen to have a $k/2$-dimensional
section which is a multiple of an Euclidean ball.
Can…
J. Dianous
- 87
4
votes
2 answers
Is duality of convex bodies in itself a convex function?
Let $K,T\subset\mathbb{R}^{n}$ be convex bodies (i.e. nonempty, compact and convex) containing the origin, and let $\lambda\in(0,1)$.
Fix $\langle \cdot ,\cdot \rangle$ to be the standard dot product in $\mathbb{R}^n$ (though probably here any dot…
Donjim
- 43