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The problem has been studied for various groups $G$ of isometries of $\mathbb R^n$. A set $K\subset \mathbb R^n$ is called $G$-universal cover iff every set of diameter 1 is contained in $gK$ for som …

This is a very old problem and there is a classical analytic approach to it. You can express the volume of sections of a convex body in terms of the Fourier transform of powers of the Minkowski functi …

This is a huge subject. The minimum sizes of $\epsilon$-nets of compacts in linear spaces were studied by Kolmogorov and his school. They showed that in general there are no good bounds for this quan …

There is a general version of this question which is known as "the rumpled dollar problem". It was posed by V.I. Arnold at his seminar in 1956. It appears as the very first problem in "Arnold's Prob …

Assume that $A$ is compact and convex. If there is a point $P$ such that any line through it is a bisector of $A$ then $A$ has to be centrally symmetric. In fact a stronger result is known (see the pa …

Fillmore showed that there are sets of constant width in $\mathbb R^d$ with analytic boundaries which have a trivial symmetry group (so these are very different from spheres; see "Symmetries of surfac …

I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for. The volume of an $n$-dimensional ball of radius $R$ is given by the classical f …

The short answer is that there are no particular constraints on the spectral decomposition of the function $\varphi$, as long as a basic convexity condition is satisfied. Lemma..Assume that $\var …

The rolling motion of a convex symmetric body on a horizontal plane is a classical problem. In the symmetric case, Chaplygin was the first who showed that the full equations of motion can be reduced t …

Whereas I don't know of any recent progress in this problem, let me mention one result for closed curves. Theorem. A closed plane curve of length $L$ and curvature bounded by $K$ can be contained …

This problem has been studied by many people in a general setting of convex bodies. Given a convex body $K$ in $\mathbb R^d$ and an $\alpha > 0$, find the maximal number $H_{\alpha}(K)$ of nonover …

There is a sharpened version of the plane isoperimetric inequality due to Benson which involves the inner and outer radii. Let $$\Gamma=\{(r,\theta):\ r=r(s),\theta=\theta(s)\}$$ be a simple closed re …

Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume? Background/motivation. A 2D version of the …

E. Torricelli, who was the last Galileo's secretary, suggested a purely geometrical method to find the envelope in his De motu Proiectorum. He also coined the term `parabola of safety'. Apparently i …

You might be interested in the recent article by Böröczky and Csikós concerning polytopes $P_n$ of $n$ facets with minimal surface area. They studied best approximations of a convex body $K\subset \ …