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It is well known that any ray of light passing thru a focus of an ellipse will pass thru the other focus after a single reflection from the ellipse boundary. If $A$ and $B$ are the foci of an ellipse, …

We continue from Cutting convex regions into equal diameter and equal least width pieces - 2 Question: If a planar convex region C is to be cut into n convex pieces such that the average of the perime …

Ref: Mathematical Omnibus by Fuchs and Tabachnikov, Lecture 11. Consider any planar convex region C. A line l on the same plane and cutting thru C may be called an inertia bisector of C if it divides …

We add a bit to A claim on the concurrency of area bisectors of planar convex regions Define a width of a planar convex region $C$ as the distance between two parallel lines that just touch $C$. A wid …

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 …

This is an extension of On segments of equal area cut from planar convex regions by chords. While the 3D analog of the above question would be about 3D pieces cut from a convex body $C$ by planes, her …

This question is closely related to Convex polyhedra with non-congruent faces It is known that if all faces of a tetrahedron ought to have same area (or same perimeter), then, the faces are necessaril …

We try to proceed from Least area and least perimeter triangles that contain a convex planar region - how different can they be? The partial answer given to the above question shows a convex quadrilat …

This post presents a variant of On segments of equal area cut from planar convex regions by chords Consider a planar convex region C of unit area and all chords of it that cut off a segment of area α …

This is a constrained version of the 'fair partition' ('spicy chicken' - https://arxiv.org/abs/1306.2741) question. It seems that there are convex polyhedrons that cannot be cut into n convex pieces …

A planar region C such that there is an interior point that bisects all chords of C that passes through it may be termed centrally symmetric. It appears that such figures exist in non-Euclidean geomet …

Question: If within a convex solid body C there is a special point P such that every planar section of C passing through P has the same area, then, can we assert that C is a sphere and P its center? I …

Given a convex polyhedron, one considers 3 possibilities: wireframe - only the edges of the polyhedron have mass which is uniformly distributed. surface - only the surface is massive with uniform dis …

Ref: A center of convex planar regions based on chords The above discussion quotes the definition of 'centralness coefficient' and defines a center of a planar convex region. 1/2 is the least possible …

Given a convex polygonal region C, how does one find/characterize the smallest zonogon (centrally symmetric convex polygon https://en.wikipedia.org/wiki/Zonogon) that contains C? Note 1: In question …