# Tagged Questions

**5**

votes

**1**answer

99 views

### References to proofs of a theorem by Van Kampen-Flores

Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other.
This ...

**2**

votes

**1**answer

103 views

### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...

**6**

votes

**1**answer

94 views

### Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant ...

**2**

votes

**1**answer

101 views

### Circumscribed ellipsoid of minimum Hilbert-Schmidt norm

Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ...

**6**

votes

**2**answers

230 views

### Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?
I found only one reference, to "Über einige ...

**3**

votes

**0**answers

70 views

### Covering points with a convex hull

Consider a set of $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$, for some $n \gg d$. Suppose $\{x_1,\ldots,x_n\} \subset C \subset \mathbb{R}^d$. Say that a set of points $y_1,\ldots,y_m \in C$ covers ...

**4**

votes

**1**answer

241 views

### A question of compactness in the geometry of numbers

Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the critical determinant of $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices ...

**11**

votes

**1**answer

307 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**9**

votes

**1**answer

725 views

### Reference request: Ehrhart's conjecture on the geometry of numbers

Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n ...

**7**

votes

**3**answers

242 views

### Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$

Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...

**4**

votes

**4**answers

801 views

### Delaunay triangulations and convex hulls

This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...

**5**

votes

**1**answer

310 views

### Isometric embedding a convex cap to render its boundary planar

I would like to know if there is a polyhedral analog to this beautiful
theorem of Hong:
Theorem 11.0.1.
Any smooth positive disk $(\bar{D},g)$ with a positive geodesic
curvature along ...

**10**

votes

**2**answers

902 views

### The cone of positive semidefinite matrices is self-dual? (reference needed)

I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long ...

**12**

votes

**3**answers

595 views

### Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...

**10**

votes

**2**answers

556 views

### Helly theorem + Nerve

Consider nerve $\mathcal N$ of a finite set of convex sets in $\mathbb R^n$.
Helly theorem says that $\mathcal N$ is completely determined by its $n$-skeleton, say $\mathcal N_n$.
It seems that not ...

**3**

votes

**0**answers

183 views

### Maximum surface area among convex subsets of the unit sphere of a given volume

The following problem is listed in Steven Lay's "Convex Sets and Their Applications" (1982) as unsolved (paraphrased):
Let $B$ be the unit ball in $\mathbb{R}^3$ and $0 < V < \pi$. Define ...

**3**

votes

**0**answers

120 views

### Dimension of convex arrangements for hypergraphs

Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A.
Has ...

**7**

votes

**1**answer

426 views

### A variation of Minkowski sum

I have to work with the following variation of Minkowski sum:
Let $\mathbb E$ be a Euclidean space and $K$ be a convex set in $\mathbb E\times \mathbb E$.
Set
$$K^+=\{\\,x+y\in\mathbb ...