# Questions tagged [discrete-geometry]

Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

**2**

votes

**1**answer

39 views

### Probabilities of Four Points Being in Convex/Deltoid Configurations

Question:
what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum ...

**2**

votes

**1**answer

60 views

### Diagonal shortcuts to minimize all-pairs shortest-paths in grid graph

Augment the grid graph $G$ on lattice points $[1,n]^2$, which
connects each point to its four distance-$1$ vertical and horizontal neighbors.
Augment $G$ to $G'$ by adding in one of the two $\sqrt{2}$ ...

**4**

votes

**0**answers

69 views

### Discrepancy of the finite approximation of the Lebesgue measure

Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...

**6**

votes

**2**answers

105 views

### Geometric dissection theory

A few days ago, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of ...

**3**

votes

**1**answer

136 views

### Sphere packing and kissing numbers in 3D

When one looks at the way cannon balls and oranges are normally packed by the military and by groceries, it seems intuively clear that there is no way anybody can pack these any tighter. However, it ...

**2**

votes

**0**answers

43 views

### Optimal $f$-vector properties of translationally invariant 3-honeycombs for error correction of a photonic quantum computer

In terms of the $f$-vector for a translationally invariant (in $\Bbb R^3$) honeycomb define
$$
\begin{split}
v &= \max\left( \frac{f_1}{f_0}, \frac{f_2}{f_3}\right), \\
f &= \max\left( \frac{...

**2**

votes

**1**answer

99 views

### Number of bitangents to convex polytopes

Let me state my question prior to defining terms:
Q. Let $P_1$ and $P_2$ be disjoint convex polytopes
in $\mathbb{R}^d$ of $n$ vertices each.
What is the maximum number of distinct bitangent
...

**5**

votes

**0**answers

58 views

### Minimize number of lattice paths below a given path

Every north-east lattice path (NE-path) $v$ from $(0,0)$ to $(k, a)$ can be identified with a sequence $0 \le \lambda_1 \le \lambda_2 \le . . . \le \lambda_k\le a$, that represent the hight of each ...

**3**

votes

**0**answers

94 views

### Are triangulations with common refinements PL-homeomorphic?

Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...

**3**

votes

**1**answer

57 views

### Tiling the surface of a hypersphere with regular simplices

Let $S^{n-1} = \{x \in \mathbb{R}^n : x_1^2 + \cdots + x_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p_1, \...

**3**

votes

**1**answer

248 views

### How to find the vertices of the set $\{v_i\in \mathbb{R}:a_1\ge v_1\ge v_2\ge \cdots\ge v_n\ge 0,\ q_2\le \sum_{i=1}^n p_iv_i\le q_1\}$

I am given a set of inequalities $v_1\ge v_2\ge \cdots\ge v_n\ge 0$, $q_2\le \sum_{i=1}^n p_iv_i\le q_1$, with $\{p_i\}_{i=1}^n,\ q_1,q_2$ positive reals, and only one bound for the coordinates: $v_1\...

**4**

votes

**1**answer

223 views

### Graphs with adjacency matrix depending on associated-vector distances

Let $G$ be a graph of order $n$ such that for each vertex $v$ there are two associated vectors, $f_v, g_v\in R^n$, where $uv\in E(G)$ if and only if $\|f_u - f_v\|^2 \ge \|g_u-g_v\|^2$.
ISGCI didn't ...

**1**

vote

**0**answers

113 views

### Is there a method to cut a hypercube into disjoint cubes [closed]

Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$
so we can cut a hypercube into at least $n+1$ disjoint parts.
Is there a method how can one do that?

**2**

votes

**0**answers

79 views

### A theory of (or reference for) symmetric point arrangements

I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...

**0**

votes

**1**answer

116 views

### Reference request on Borsuk conjecture [closed]

I just heard of Borsuk conjecture. I want to ask if there are any references preferably looking at the problem from the point of view of Mathematical analysis I can study it from?
Thanks