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This question was explored here: Lenhart, William J., and Sue H. Whitesides. "Reconfiguring closed polygonal chains in Euclidean $d$-space." Discrete & Computational Geometry 13, no. 1 (1995): 123 …

Permit me (as a late-comer) to add a bit more information. The 2D version of your question was posed by R. Nandakumar and N. Ramana Rao and posted at TOPP. They have written "an introduction" to thei …

Let $A$ be the expected area. Then: $$\lim_{n \rightarrow \infty} \frac{n}{\ln n} (1 - A) = \frac{8}{3} \;.$$ This can be found in many places, e.g., this MathWorld article. [Updated with comparison …

I am posting this on behalf of Costin Vîlcu (with whom I've had the pleasure of coauthoring). —J.O'Rourke The posted problem in $\mathbb{R}^3$ was answered in: Y. G. Nikonorov and Y. V. Nikonorov …

OP: "I'm curious about more recent research in this area." Here are two relatively recent papers. Ivan visits MO, so he may answer more definitively. Izmestiev, Ivan. "Infinitesimal rigidity of co …

Not an answer, just an illustration to accompany the question. $K$ is an isosceles triangle with base $2$ and altitude $3$ (and so area $3$). First, I mistakenly computed the ellipse $E$ of any area w …

I believe this is an open problem. This 2014 paper Dumitrescu, Adrian, Sariel Har-Peled, and Csaba D. Tóth. "Minimum convex partitions and maximum empty polytopes." Scandinavian Workshop on Algori …

I don't know if this is the optimal, but an isosceles triangle with base and height $\sqrt{2}$ overlaps $2 \left(\sqrt{2}-1\right) \approx 0.828427$ when placed as below, and so improves over $\frac{ …

I believe you are looking for the radius of a largest empty ball among your point set, a quantity which goes under the name of dispersion. This plays a role in robotics algorithms, e.g., LaValle's boo …

To follow up on Dirk's observation in the comments, here is a smoothed version of a Reuleaux triangle with $s(x)=c$, as illustrated by the dashed normal chords, which each pass through a corner of th …

Here is an image of the optimal open convex curve. Taken from Open Problems from CCCG 2012, based on this paper, which cites Nudel'man (1975): Paolo Tilli. "Isoperimetric inequalities for conve …

If you search for detection of redundant constraints in linear programming you will find many hits, including one to an MO question, "Detection of Redundant Constraints." One source paper is J. Go …

There exist convex quadrilaterals which have no such splitting. And there is an $O(n^3)$ algorithm to decide if such a splitting exists for a (nonconvex) $n$-gon. See the paper by Dania El-Khechen, Th …

Not an answer, just a thought. It seems a bit too much to hope for. Here are 300 points randomly sprinkled on the surface of an ellipsoid in $\mathbb{R}^3$, and then projected to two dimensions. No m …

Helly's theorem plays a role in economics theory and in game theory (noncooperative games): Fuchs-Seliger, Susanne. "An application of Helly's theorem to preference-generated choice corresponde …