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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 …

If I may add one explicit example as a late-comer, even though it is along the lines with which you are already familiar: Cauchy's arm lemma may be used to prove that the curve that is the intersectio …

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 …

The paper by Chazelle and Liu, "Lower bounds for intersection searching and fractional cascading in higher dimension." 2001. (ACM link) shows that for planar convex subdivisions, the answer …

You might see if this paper helps: Ranestad, Kristian, and Bernd Sturmfels. "On the convex hull of a space curve." arXiv:0912.2986 (2009). (arXiv abstract link)             The yellow surface …

An attempt to illustrate Yoav's construction:             (I'm responsible for any misinterpretations.)

(This is not an answer, just an illustration.) Following on Manfred's mention of oloids, I thought I would show at least one example where the OP's union-hull equation holds: $A$ and $B$ are planar s …

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 …

Not an answer, just an illustration.                     Three $2 \times 1$ ellipses, rotated $30^\circ$, reflected through different points. Hulls have the same area.

Just a historical footnote. The first proof appeared in a paper by Freeman & Shapira: Freeman, Herbert, and Ruth Shapira. "Determining the minimum-area encasing rectangle for an arbitrary closed c …

This is not a direct answer to your request for lower bounds; just some remarks. Sometimes the center of the largest enclosed ball is called the Chebyshev center, and you can find literature under t …

It is NP-hard to compute the minimum volume enclosing ellipsoid of a set of points (if the dimension is part of the input). There have been efficient approximation algorithms developed, e.g., P. K …

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 …

Grows as $O(\log^{n−1} N)$. Below, your $n$ is H-P's $d$, and your $N$ is H-P's $n$. So: $O(\log^{d−1} n)$, or, to avoid notational ambiguity: $O((\log n)^{d-1})$ Har-Peled, Sariel. "On the expected …

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 …