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For every convex compact set $K$ of area $1$ in $\mathbb{R}^2$, among all ellipses of area $1$ there exists an ellipse $E$ such that the area of the symmetric difference between $K$ and $E$ is smalles …

Can every 3-dimensional convex body be trapped in a tetrahedral cage? Although the question is fairly unambiguous, I give all relevant definitions: $\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-d …

How large can the intersection of a parallelogram (or a square, if you prefer) with a triangle be, if each of them is of unit area? It is easy to see that the intersection can be of area 3/4 – is this …

Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all dir …

Let $K$ be a convex body of volume 1 in $\mathbb{R}^n$ and $x$ a (variable) point on the boundary of $K$. Define $f_K(x)$ to be the volume of the convex hull of the union of $K$ with its reflection in …

This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of …

The diameter of a bounded set is the supremum of the distances between any two points of the set, and the circumradius is the infimum of the radii of balls containing the set. Obviously, the diameter …

This question is related to How to make a sandwich from just one piece of bread?, asked on Feb 23 '17 by erz, and it goes as follows: Question. If a convex closed and bounded region $C$ in the pl …