# 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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I was reading Adam Sheffer's book "Polynomial Methods and Incidence Theory" and I tried to solve the following exercise: Exercise 1.1 Construct a set $\mathcal{P}$ of $m$ points and a set $\... 0 votes 0 answers 78 views ### On smallest convex m-gons that contain a given n-gon where m<n Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ... 1 vote 0 answers 46 views ### On points in the interior of planar convex regions and inscribed triangles Given any planar convex region C, it is easy to show that every point in the interior C is the mid point of at least one chord of C. Likewise, Question: Is every point in the interior of C the ... 1 vote 0 answers 37 views ### Locality and restriction properties for self-avoiding and loop-erasing random walks This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa I ... 11 votes 1 answer 497 views ### A variant of the corners problem Question: What is the size of the largest subset of$[n]^2$containing no three point configurations of the form$(x,y), (x,y+d), (x+d,y')$with$d \neq 0$? In particular, is it at most$O(n)$? Recall ... 1 vote 1 answer 65 views ### When do the centers of mass of a uniform convex planar region as a whole and of its boundary alone coincide? Given a uniform planar convex region C, let us consider 2 centers of mass - the center of mass of the region as a whole and the center of mass of its boundary alone (assuming its boundary to have ... 0 votes 0 answers 135 views ### Equivalent formulation of Szemerédi-Trotter theorem I am reading the first chapter of Adam Sheffer's book "Polynomial Methods and Incidence Theory" and in Lemma 1.15 he proves an equivalent formulation of Szemerédi-Trotter theorem. Before ... 6 votes 0 answers 118 views ### Have the affine simplicial line arrangments been enumerated? I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements. A line arrangment is a family of straight lines in$\Bbb R^2$. It is simplicial if all regions are ... 8 votes 2 answers 464 views ### Continuous point map for spherical domains Consider the space$J$of Jordan domains on the sphere$\textbf{S}^2$, i.e., continuous injective maps from the unit disk into$\textbf{S}^2$modulo homeomorphisms of the disk. How can one construct a ... 2 votes 1 answer 122 views ### Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle We say a rectangle has orientation$\theta$if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle$\theta$with X axis. Consider a planar ... 1 vote 1 answer 72 views ### To optimally wrap convex laminae with paper Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both ... 4 votes 0 answers 196 views ### What does it mean "parallel"? I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following Definition 1: An affine set$A$is parallel to an affine set$B$in a linear ... 8 votes 3 answers 718 views ### Alternating Sum Involving Catalan Numbers I was wondering if anyone knew how to obtain a simpler closed form of the following sum(or had any other insights regarding it): $$\sum_{k=0}^n (-1)^k{n \choose k} C_{2n-2-k}$$ Here$C_n = \frac{1}{n+...
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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 ...