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The classical electric current density can be modelled as a 2-form $$J=J_{ij}\wedge dx^{ij}$$ which is assumed to be locally integrable over a 3-manifold (3-dimensional domain) $X$. By integrating $J$ …

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 …

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 …

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 …

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 …