# Questions tagged [discrete-geometry]

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

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### Does a matroid base polytope contain its circumcenter?

Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x_1,...,x_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its matroid base polytope (i.e. the convex hull of the ...
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### Which rectangles can be cut into finitely many rectangles all with same perimeter and different areas?

Ref 1: dividing a square into unique rectangles with the same perimeter https://arxiv.org/ftp/arxiv/papers/1307/1307.3472.pdf Ref 1 asks if a square can be cut into some finite number of rectangles ...
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### Convex 3d bodies for which all planar sections with max diameter have same diameter

Ref: 1. A claim on planar sections of 3D convex bodies On convex 3d bodies whose shadows are all of constant diameter Given a 3D convex body $C$ and a specified direction $n$, we consider the planar ...
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### On convex 3d bodies whose shadows are all of constant diameter [closed]

We add a bit to More on shadows of 3D convex bodies By a shadow of a 3D body, we mean the orthogonal projection of it onto a 2D plane. If all shadows of a convex 3D body have the same diameter, will ...
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### A projective plane in the Euclidean plane

Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is ...
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### On largest convex m-gons contained in a given convex n-gon where m < n

This post is the inside-out variant of On smallest convex m-gons that contain a given n-gon where m<n Given a convex n-gon region P, and an m less than n, how to find the max area convex m-gon Q ...
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### Membership test of convex set

Let $K$ be a compact convex subset of $R^n$ which has some positive gaussian measure, say at least 1/2. For each nonzero vector $u \in R^n$, we define another compact convex set $K * u$ in the ...
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