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(i) It is known (Zalcman, 1982) that the Radon transform is not injective: there exist non-trivial continuous functions $f\colon \R^2\to \R$ with zero (proper) integral on every (doubly infinite, straight) line. All known examples of such functions have extremely rapid overall growth. Can such a function have slow growth, or even be bounded? Is it true that a continuous function on $\R^3$ with zero integral on every line must be identically zero?

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(ii) Every polygonal domain $D$ in $\R^2$ has the Pompeiu property (PP): if $f\colon \R^2\to \R$ is continuous and $\int_{\sigma

(D)}f(x)dx =0$ for every rigid motion $\sigma$, then $f\equiv 0$. The corresponding assertion for functions on the sphere $S^2$ is

false: there are infinitely many (non-congruent) regular spherical polygons that lack PP, and they can be characterised. But it still

seems unclear whether, for example, all non-trivial regular spherical triangles have PP, and whether (up to congruence) the

known example of a spherical square lacking PP is unique.

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(iii) One formulation of the maximum principle asserts that if $h$ is a non-constant harmonic function on a ball centred at the

origin $O$ in $\R^n$ and $h(O)=0$, then $h$ takes positive values and negative values on every neighbourhood of $O$. In the case

$n=2$, this can be quantified: it is easy to show that, with $h$ as above, the subset of $\{x\colon ||x||<r\}$ where $h>0$ and the

subset where $h<0$ have roughly the same area. (The ratio of the areas tends to 1 as $r\to 0+$.) What can be said in the case $n\ge 3$?