# Tagged Questions

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

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### Do the last three remaining cards in a game of Set always form a set?

Question (brief introduction to the game Set is given after the question) When a game of Set gets to a point that there are only three cards left on the table, and all other cards were already ...
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### Sufficiently many points in $\mathbb{R}^d$ must contain $m$ points forming the vertices of a convex polytope?

Let us say that a set of points in $\mathbb{R}^d$ is minimal if it forms exactly the set of vertices of a convex polytope. Equivalently, no proper subset of the points has the same convex hull; no ...
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### How Many Triangles are Created by n Lines in the Plane?

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following: A. no 2 lines are parallel or identical B. no 3 lines have common ...
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### Grid with both squares and equilateral triangles

Is it possible to have a grid that contains both squares and equilateral triangles? By grid I mean any set of the form $M \mathbb Z^2$, with $M \in GL_2\mathbb R$. I think this is impossible, ...
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### What's the upper bound for sofa problem?

I have seen a claim that for the sofa problem, an upper bound for the area of a sofa is $2 \sqrt 2$, and that this can be proved by a "simple" argument. But I can't find a proof. What that argument? (...
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### Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
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### Plane tesselation, using stairs $n\times n$, is it possible?

The other day I was constructing new mathematical problems for my pupils and thought of something like this: Given the infinite sequence of "stairs" $n\times n$, constructed from $1\times1$ ...
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### Upper bound on the minimum distance between $N$ points chosen inside the unit circle?

I guess this is a well-known problem but I'm not sure where to find it on the web. $N \ge 2$ points are chosen in the interior or the boundary of the unit circle. What is the best upper bound on the ...
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### Convergence of Discretized Geodesics?

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto \mathbb{R}^D$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the ...
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### Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm ...
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### Geometric median (or Fertmat-Webber problem), including continuous case

For a finite set $X\subset \mathbb R^n$ the geometric median is defined as the point in $\mathbb R^n$ for which the sum of distances to all points of $X$ attains its minimum. Here is a wiki article: ...
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### Projective Geometry: Combinatorially, but not projectively equivalent polytopes

I have a hard time understanding Projective Geometry. My task is to Find two polytopes, that are combinatorially, but not projectively equivalent. What combinatorially equivalent means is ...
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### Gaussian blur over (or random walk in) a surface mesh

Let $V$ be the set of mesh vertices, connected by edges $E$, forming a mesh that represents a surface embedded in $\mathbb{R}^3$. On this mesh a function $f:V\rightarrow\mathbb{R}$ is defined. For ...
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### How many acute triangles can be formed by 100 points in a plane?

Given 100 points in the plane, no three of which are on the same line, consider all triangles that have all their vertices chosen from the 100 given points. Prove that at most 70% of those triangles ...
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### On partitioning a finite set of points in the plane by drawing a line

Number of ways to separate $n$ points in the plane The linked answer was to this question: Suppose you have $n$ points in the plane, no three of which are colinear. How do you show that the number ...
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### Textbook Recommentation: Discrete Differential Geometry

are there any good books that provide a good introduction to Discrete Differential Geometry to beginners? Thanks a lot.
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### Lebesgue covering problem's demand for convexivity

Is there a specific reason why in the Lebesgue (universal) covering problem only convex sets are admitted as universal coverings? I see that for any set $X\subseteq{\bf R}^2$ of diameter $\le\alpha$, ...
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### Given two sets of points in the plane, there exists a point with equal sum of distances to the points in each set

Let ${A_1, ..., A_n}$ and ${B_1, ..., B_n}$ be two given sets of points in a plane with different centroids. Prove that there exists a point $P$ in the plane such that $\sum |PA_i| = \sum |PB_i|$. ...
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### Circle enclosing all but one of $n$ points

It looks like a simple question but it turns out rather difficult to me. Here is the question: Given $n$ points on the plane, can we always draw a circle that includes exactly $n-1$ of them?
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### Shelling of a polytope

During line shelling of a convex polytope in d-dimension, it is easy to see that visible facets are shellable. In the same way non visible facets are also shellable. But while combining these two part,...
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### Largest rectangle not touching any rock in a square field

You want to build a rectangular house with a maximal area. You are offered a square field of area 1, on which you plan to build the house. The problem is, there are $n$ rocks scattered in unknown ...
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### Polygons with a Unique Triangulation

For each n > 3, find a polygon with n vertices that has a unique triangulation. I want to say that you can somehow build these polygons by continuously adding triangles somehow, but I'm not sure.
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### Elementary proof of Jordan curve theorem for polygons

Courant described the outline of an elementary proof of the Jordan curve theorem for polygons using the order of points: The order of a point $p_0$ is defined by the net number of complete ...
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### What is the optimal tiling of a regular n-gon in the plane?

I want to tile the plane with equal-sized regular polygons of $n$ sides. Obviously for some $n$, the tiles will be able to tessellate and cover the whole plane (e.g triangles, squares, hexagons) I ...
Define the unit ball centered at the origin as $B=\{x\in\mathbb{R}^d\mid \|x\|\leq 1\}$. Define the dual set of set $X$ as $X^*=\{y\in\mathbb{R}^d\mid\langle x,y \rangle\leq 1\ \forall x\in X\}$. I'...