As a discrete analog of the MO question, "Löwner-John Ellipsoid: incribed and circumscribed," I've been wondering what might be the maximum ratio of this quantity? Let $P$ be a convex polygon of $n$ vertices, $P^+$ be a minimum area polygon of $n{-}1$ vertices circumscribing $P$, and $P^-$ a maximum area polygon of $n{-}1$ vertices inscribed in $P$.
Over all polygons $P$ of $n$ vertices, what is the maximum of the ratio area$(P^+)/$area$(P^-)$, as a function of $n > 3$\ge 4$?
Here are possible optimal (en/in)closures for a square and a regular pentagon:
[Some updates below.]
I don't know for certain that any of the illustrated polygons are optimal.
I believe that it is known that the circumscribing $(n{-}1)$-gon must have one
edge "flush" with $P$ (i.e., including an edge of $P$ as a subset), and
that this may be proved in a 1984 paper by
Chang and Yap
"A polynomial solution for potato-peeling problem"
(Discrete & Computational Geometry
Volume 1, Number 1, 155-182), which I cannot access at the moment.
This is certainly true for enclosing triangles.
I am not sure if there is any analogous characterization of inscribed $(n{-}1)$-gons.
A key result for minimum area circumscribing is by Victor Klee in 1986:
"Facet-centroids and volume minimization,"
Studia Scientiarum Mathematicarum Hungarica, Vol. 21, 143-147, 1986.
As the title indicates, he proved that each facet's centroid must touch $P$ (in any dimension).
The circumscribing polygons I drew above have their edge midpoints touching $P$.
For $n{=}4$, $a \times b$ rectangles have an area ratio of $(2 a b) / (a b / 2) = 4$, illustrated with the square above; perhaps this is the worst ratio over all $n$?
Any ideas would be appreciated, from a clean proof (or counterexample!) that one circumcribing edge must be flush, to any constraints on the inscribed polygons, to an answer to the ratio question, even for specific $n$, even for regular $n$-gons. Thanks!
