# Questions tagged [discrete-geometry]

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

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### Is a polytope that has in-spheres for faces of all dimensions already regular?

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...
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### How much smaller is the Čech complex than the Vietoris-Rips complex?

The Čech complex is a subcomplex of the Vietoris-Rips complex. The V-R complex includes as a simplex a set of points with pairwise distances at most $\epsilon$, whereas the Č complex includes as a ...
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### Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?

Is the following lemma a well known result in graph theory? I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
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### What does the extension theorem for tilings state?

I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space. E.g. in "The Local Theorem for Monotypic Tilings" one reads The Extension Theorem [......
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### Reference request: placing a set with respect to the integer grid

For $x=(x_1,...,x_n)\in \mathbb{R}^n$, let $Q_x=(x_1,x_1+1)\times ...\times (x_n,x_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft here) the following ...
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### Which convex bodies can be captured in a knot?

Which convex bodies can be captured in a knot? This question is based on the discussion in "Is it possible to capture a sphere in a knot?". We assume that the knot is made from unstretchable, ...
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### How many Shapes are possible to create using Voxels?

Let's suppose I have Big Cube of x cm by y cm by z cm, simmilar to this one: This big cube is made of tiny little cubes of t cm. All of this little cubes are transparent, but some of them are red ...
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The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$. The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\... 2answers 169 views ### Maximal distance of$2d+1$points on a sphere If one arranges$2d$points on the sphere$\mathbf S^{d-1}\subset\Bbb R^d$as the vertices of the regular octahedron, then one can achieve a minimal spherical distance of$\pi/2$between any two ... 2answers 194 views ### A question about dense sets Suppose that$A$is a given subset of$I=[0,1],\ $and$ \left\{ x_j = \frac{j}{m} \right\}_{j=0}^{m}\ $is the$m$-partition of$I$, and$\nu(m)$is the number of$\ [x_{i-1},x_{i}]\ $such that$\ [...
Let $P\subset\Bbb R^d$ be a convex, full-dimensional polytope (convex hull of finitely many points, affine hull is the whole space), $G_P$ its edge graph and $\mathcal F_P$ its face lattice. Any of ...