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Your Question 2 (max/min area/volume shadow version) is answered in this paper: McKenna, Michael, and Raimund Seidel. "Finding the optimal shadows of a convex polytope." In Proceedings 1st Annual Sym …

The left shape below has $3$ contacts (circled) "between pairs of units" and hull area $> 3$, while the right shape has $2$ contacts and area $3$. So minimizing the hull area does not always maximize …

Here is an illustration of Gerry Myerson's nice idea: The left set has onion depth $n/3$, the right set, after small rotations, has depth 1. Incidentally, there is an efficient algorithm to find the …

Let $C$ be a cone with lateral side $A$ and base $B$. So $C = A \cup B$. The base is geodetically convex. But if I'm interpreting the definition correctly, $A$ is not geodetically convex: For any two …

Not an answer; just an illustration. I had some difficulty understanding the question, so... Here $n=2$, so the shapes are planar, $\mathbb{R}^2$. I used Reuleaux triangles for the $3=n{+}1$ unit-cons …

Pardon me for this bit of self-promotion, especially because this is only tangential to the OP's concerns. But cone unrolling and Anton Petrunin's mention of Alexandrov's developments, in conjunction …

These references may help? Or at least lead you to related literature. João Gouveia and Rekha Thomas. "Convex hulls of algebraic sets." In Handbook on Semidefinite, Conic and Polynomial Optimization, …

My answer here may help, especially the citation to Nie, Jiawang, Pablo A. Parrilo, and Bernd Sturmfels. "Semidefinite representation of the $k$-ellipse." In Algorithms in Algebraic Geometry, pp. 117 …

Addressing the OP's "Note," as far as I know, this is the algorithm status: There are fast approximation algorithms, but I have not found an exact algorithm (except when only translations are permitte …

Let me just quickly remark on one embedded question (but not your main questions): "then it could be called the 'area partition center' of the region and finding this center for a general given regio …

I believe this answers (1). $P$ is the pyramid illustrated. $S$ is a square resting on the apex of $P$, at height $z_1$. Projecting $S$ down (green lines) onto $P$ results in the nonconvex shape outli …

This is not an answer, and not even that helpful, but I wanted to see the central pattern formed by the collection of perimeter bisectors.                    

Perhaps it is worth quoting this theorem, even though it does not distinguish bounded from unbounded cells, and is phrased in terms of the number of hyperplanes rather than the number of points determ …

The example below seems to suggest No for the inscribed question as well. The line of symmetry is horizontal (dashed). It seems the best aligned isosceles triangle (pink) has area $A_1=\frac{1}{2} ( …

There is a notion called the "orthogonal convex hull," or the "digital convex hull," which may be what you seek. For example, in this paper, Karmakar, Nilanjana, and Arindam Biswas. "Construction …