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Questions tagged [discrete-geometry]

Discrete geometry includes the study of covering, illumination, packing, convex bodies, convex polytopes, and other metric geometry.

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

733 questions
11
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A multi-dimensional Frobenius problem

Inspired by this question. Let $A$ be a subset of ${\mathbb Z}^d$ that generates the whole additive group ${\mathbb Z}^d$, and let $S$ be the additive semigroup generated by $A$. Prove that there is a $d$-dimensional convex cone $C\subset {\mathbb…
user940
7
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1 answer

How close do distinct distances to $0$ determined by a square integer lattice in $\mathbb{R}^2$ get?

Recently on MSE's chat, user "Simd" raised the following problem (I have rephrased and introduced some notation): For $n \geq 1$ let $S_n \subseteq \mathbb{R}^2$ denote the Cartesian product of $\{0,1,2,\dots,n\}$ with itself, and let $\|\cdot\|$…
leslie townes
  • 8,842
6
votes
1 answer

Covered 10x10 rectangle with L-shapes trominos

We have given L-shaped trominos and a square of size 10x10. Give a nice proof, that 18 L-trominos is the minimal number with which the square can be covered such that it is impossible to insert one more L-tromino. I give an example with 18…
andrew
  • 131
4
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1 answer

Simple geometry problem on distribution of points in a plane

Consider 6 distinct points in a plane. Let $m$ and $M$ be the minimum and maximum distances between any pair of points. Show that $M/m \ge \sqrt3$. I am more interested in arrangement of these points where the equality holds. I figured out that the…
user67773
4
votes
1 answer

Area covered by an ellipsis on a cell grid

The following situation: if I place an ellipse of which I know the semi-axes, the orientation and the location of the centre on a cell grid, how can I figure out how much area of that ellipse falls onto which cell? The cells are all quadratic and…
DominikR
  • 43
4
votes
3 answers

Euclidean Tilings that are Uniform but not Vertex-Transitive

Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra. A…
Jamie Banks
  • 12,942
4
votes
1 answer

Set of diameter < 1 is contained in a disc of radius $\frac{1}{\sqrt 3}$

Exercise from Matousek Lectures in Discrete Geometry Prove that each set $X \subset \mathbb{R}^2$ of diameter at most 1 (i.e., any two points have distance at most 1) is contained in some disc of radius $\frac{1}{\sqrt 3}$ I'm just interested in…
Ainsley Pullen
  • 297
3
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Polytopes-Discrete Geometry

Can someone help me solve the following question please? Let v be a vertex of a d-polytope P such that $ 0 \in intP $ . Prove that $ P^{*} \cap \{ y \in \mathbb{R}^d \mid\left < y, v\right>=1\ \} $ is a facet of $P^{*} $. The definitions…
joshua
  • 1,269
3
votes
2 answers

Find center of circle of radius $r$ that overlaps exactly $\lfloor \pi r^2 \rceil$ points of the integer grid

Can a circle of a given radius $r$ always be placed (in $\mathbb{R}^2$) such that the number of points with integer coordinates inside the circle is equal to the nearest integer of the circle's area? In other words, if we imagine a square grid with…
arne.b
  • 395
3
votes
1 answer

Intersection of line with discrete hypercubes in n-dimensional space

I am looking for a method to determine the hypercubes that intersect a line between two points in a high dimensional space. I think what I want is the supercover of a line in high dimensional space. For example, if we are working in a 5D bounded…
Del F
  • 33
3
votes
0 answers

Attempt to answer to an "unanswered" in MO

EDIT : lemma 2 is wrong, see comment (i try to see if something in the proof can be saved)This [should] answer by the negative to the question…
jcdornano
  • 351
3
votes
1 answer

Decomposing a convex quadrilateral

Problem: Let $Q$ be a convex quadrilateral which is cut into convex pieces (cells) by a finite number of lines. For any collection $(Q_i)_1^n$ of these cells, decompose $Q$ into nonoverlapping convex polygons $(R_i)_1^n$ so that $Q_i \subset R_i$…
Wesley
  • 1,567
3
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0 answers

Is it possible to arrange $p^2$ points on $p^2$ lines so that there are $p$ points on each line?

There are $p^2$ points and $p^2$ lines. Two lines have no more than one intersection point. Is it possible (for any $p$) to arrange the points so that there are $p$ points on each line? For example, for $p=2$ it is possible.
Anastasia K.
  • 120
2
votes
1 answer

Filling Ratio of Unit Sphere

Consider the unit sphere $S^n$ in ${\bf R}^{n+1}$. Consider $S(r)$, a union of $r$-balls in $S^n$ which is disjoint and that $S(r)$ has maximum area. Then define $$ c_n\doteq \lim_{r\rightarrow 0} \frac{{\rm vol}\ S(r)}{{\rm vol}\ S^n}.…
HK Lee
  • 19,964
2
votes
1 answer

halving lines through the centroid of a cyclic polygon

Let $A_1, A_2,\ldots, A_{2n}$ be $2n$ points on a circle centered at $O$ with the additional property that the centroid of this set of points coincides with $O$. In other words, the sum of the vectors $OA_1$, $OA_2,\ldots OA_{2n}$ is zero. Prove or…
Dan Ismailescu
  • 1,193
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