I'm looking for any known exact results about inscribing simple convex bodies inside a convex polytope. The most famous is the [Löwner-John ellipsoid](https://en.wikipedia.org/wiki/John_ellipsoid), but as far as I understood in general there is no closed form solution to this problem. Are there some particular cases, when a solution is known explicitly (maybe not necessarily an ellipsoid, a ball, for example, will do as well) under some extra conditions?


Thank you in advance


**Update:** cancelled something stupid that I've written before



Closed form solutions for maximal subsets of convex polytopes
By Ivan's request, I describe here two algorithms for finding the maximum-size ball contained in a given $n$-dimensional convex polytope $P$. The first algorithm, requiring central symmetry of $P$, provides a solution in one step only, the second one is recursive, and may not produce the solution in finitely many steps, but yield a sequence of approximations instead, converging to a solution. Also, note that the solution is not always unique, as certain convex polytopes, even among the centrally-symmetric ones, contain infinitely many maximum-size balls. Example: any rectangle with unequal sides.


It is best to begin with the so-called **H-descripton** of $P$, as the intersection of a finite collection of half-spaces, see <https://en.wikipedia.org/wiki/Convex_polytope> and the references given there. Let $H\_1, H\_2, \ldots H\_k$ denote the boundary (hyper)planes of the half-spaces. 



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> **For centrally-symmetric polytopes:**
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Assume the origin $O$ is the center of symmetry of $P$, therefore $k$ is even and the boundary planes come in pairs reflected about $O$.
The radius of the largest ball contained in $P$ is $$ r\_{\rm max}=\min\_{1\le i\le k} \{{\rm dist}(O,H\_i)\},$$ which is easy to compute.


$\ $



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> **For non-symmetric polytopes:**
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**Step 1.** Pick a point $c\_0$ in the interior of $P$ and compute $\displaystyle r\_0=\min\_{1\le i\le k} \{{\rm dist}(c\_0,H\_i)\}$, similarly as above. The ball $B\_0$ centered at $c\_0$ and of radius $r\_0$ is contained in $P$, and it touches the boundary of $P$ at at least one but at most $k$ points on different facets of $P$. If the set $S\_0$ of points of contact between $B\_0$ and the boundary of $P$ does not lie in any open hemisphere of the boundary of $B\_0$, then the ball $B\_0$ is the largest ball contained in $P$. Otherwise do:


**Step 2.** Denote the open hemisphere of $B\_0$ that contains the set $S\_0$ by $D\_0$. Then let $E\_0$ be the hyperplane containing $c\_0$ and disjoint from $D\_0$. Now move the ball $B\_0$ perpendicularly to $E\_0$ and away from $D\_0$ to a new position $B'\_0$ by a small translation, small enough to make sure that $B'\_0$ is contained in the interior of $P$ and let $c\_1$ denote the center of $B'\_0$.


Now repeat Step 1 taking $c\_1$ in place of $c\_0$ and getting $B\_1$ and $r\_1>r\_0$ in place of $B\_0$ and $r\_0$.


Continue recursively until the set of points of contact of $B\_m$ with the boundary of $P$ does not lie in any of the open hemispheres of the boundary of $B\_m$. The ball $B\_m$ is then the maximum-size ball contained in the polytope $P$.


