I am looking for bibliography on the following problem.Given $N\in\mathbb{N}$ find $N$ points $p_1,...,p_N\in\mathbb{R}^2$ which(1) maximize $\min_{i,j} |p_i-p_j|$(2) subject to the constraint $\...

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This is somewhat related to this old question of mine, but is hopefully easier.Let $\mathcal{H}$ be the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric, and let $\...

I am looking for some examples of triangle-free graphs/1-skeletons of convex $d$-polytopes with $d\ge 4$ whose chromatic number is at least 4, specially in dimensions 4, 5, 6. I know of one 7-...

I recently tried to see in how far polyhedral geometry can be reduced to the study of convex sets with finitely many faces. In other words, I tried to replace "finitely generated" by "...

Let $S \subset \mathbb{R}^n$ be an $n$-simplex with integer vertices. Suppose that $S$ does not contain any integer point other than its vertices and that $2S$ contains at least one integer point in ...

Let $k, r \in \mathbb{N}$ with $k \ge 2$ and $r \ge 3$, and let $n_1, n_2, \ldots, n_k \in \mathbb{N}$ satisfy $3 \le n_1 \le n_2 \le \cdots \le n_k$.We define${G_n}^k := \{0,1,\ldots,n_1-1\} \times ...

There are $n$ points on the plane, satisfying that for any two points $A, B$, there is a unique point $C$ lying on the perpendicular bisector of $AB$, i.e. $CA=CB$.Prove or disprove: $n$ is odd, and ...

We add a little to On reconstructing convex polyhedrons from disconnected faces that are all mutually non congruentWe call a set of polygonal regions that all together form a convex polyhedron a ‘...

If two regular tetrahedra $S_1$ and $S_2$ in $\mathbb{R}^3$ share a triangular face $f$, we will say that each of them is obtained from the other by stacking over $f$. Now, choose some regular ...

Ref: https://arxiv.org/pdf/1307.3472It is well known that given only a set of convex polygonal regions (call this set of polygons a 'face set') and no further information, one cannot uniquely ...

Ref: To choose a set of $n$ rectangles which together form largest number of rectangular layoutsWe present a variant of above question:General Question: given an integer $n$, how do we find $n$ ...

For a triangulation $T$ of an $n$-dimensional cube, whose vertices are the $2^n$ original vertices, let $d(T)$ be the largest Hamming-distance of two vertices that are in the same simplex.How small ...

Ref 1: https://arxiv.org/pdf/1711.04504Ref 2: On 'Walls' with 'non-tiles' - a variant of the Heesch problemIt is known that we cannot tile the plane with triangles that are pairwise ...

Ref: "Non-tiles and Walls - a variant on the Heesch problem" (https://arxiv.org/pdf/1605.09203)Definitions (adapted from above doc): A non-tile is any polygon that does not tile the ...

We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar trianglesStarting with 2 unit squares and with squares with sides 2,3,5,... (all Fibonacci numbers), one can form an ...