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Convex quadrilateral decompositions of equal area are always possible. Given a convex polygon $P$ with at least $5$ sides, draw a line from one of the vertices of $P$ to a point $Q$ on one of the edge …

Not a proof, but some thoughts on how to get empirical data for this problem and evidence for the optimal configuration: Suppose we have a matrix $C$ whose rows are the unit vectors pointing in the di …

The error is in this sentence: As a circle with its centre on the Base Plane, $T_{n-1}$ will have at least two points on the Base Plane. Circles in an ambient space of more than three dimensions don …

Any member of a nontrivial family like this has to be a rep-tile; looking among those will give you many examples. Some specific examples: The family of all rectifiable polyominoes, which includes al …

(Formerly on MSE.) Here is a fairly natural question: Can three-dimensional space be filled with convex polyhedra of the same incidence structure (if not the same geometry) as the regular dodecahedron …

This is not a complete answer to all 6 questions, but provides some progress: 1. I assume that you want $C$ to be bounded here? Your language throughout the post seems to assume it. If not, then eithe …

The answer is (very likely) no for all $n>20$. Let $R_n$ be the regular $n$-gon of circumradius $1$, and given a convex shape $C$, let $\delta(C)$ be its maximal packing density in the plane. Observe …

Every such $S$ has a periodic tiling, in which finitely many disjoint copies form a set with one representative for each translate of some discrete lattice $L$ - see Bhattacharya 2016 or Greenfeld and …

The maximum area inscribed convex $m$-gon can always be found on the vertices. To do this, just consider moving one vertex at a time; we gain area the further we move this point from the diagonal conn …

Question 2 can be answered in the affirmative, at least: there are many triangles with the multi-way rep-tile property. Every triangle has a simple tiling of itself with $k^2$ copies (just take the af …

As a supplementary answer, here are some notes on the 2D analog mentioned in the original post. The conjecture is false for nonconvex bodies: The above shape can tile the plane without gaps when rota …

Yes. Place five points $P_1,P_2,P_3,P_4,P_5$ in a regular pentagon inscribed in the unit circle centered at the origin. For each of these points $P$, we're going to add another point $Q$ somewhere on …

Yes, it is possible; in fact, we can do it entirely with $5-12-13$ right triangles at different scales. First, note that we can three triangles at scales in the ratio $5:12:13$ to form a $5\times 12$ …

The 2000 paper Inequalities for Convex Sets, by Paul R. Scott and Poh Way Awyong, lists various inequalities on 2D convex bodies as described here; for your question you can see the state of the resea …

This MSE question exhibits two non-regular tetrahedra which can be decomposed into 8 smaller copies congruent to themselves; this yields $8^n$ for any $n$.