# Radon transform and Log-concavity

This question is related to (but different from) that of Darsh Ranjan.

Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat f(\omega,t)$ is log-concave in $t$ for every $\omega\in S^{n-1}$ ? Recall that $$\hat f(\omega,t)=\int_{\omega\cdot x=t}f(x)d\mathcal L_{n-1}(x),$$ with $\mathcal L_{n-1}$ the $(n-1)$-dimensional Lebesgue measure. Recall also that $g\ge0$ is log-concave if $\log g$ is concave. It is OK for me if one assumes that $f$ is integrable with compact support.

The motivation comes from a paper in collaboration with Th. Gallay, to appear in CPAM. With each $N\times N$ complex matrix $A$, we associate a numerical measure, which is a probability supported by the numerical range $W(A)\subset\mathbb C$. It is absolutey continuous with piecewise smooth density $f_A$. This density has the Radon-log-concavity property mentioned above. When $A$ is normal, $f_A$ is itself log-concave, thanks to the Prékopa-Leindler theorem. But this is no longer true for a non-normal matrix. If $N=2$, this fails already.

-
What's the connection between $N$ in the second paragraph and $n$ in the first? – John Bentin Jan 31 '11 at 16:37
@John. In the paper, $n=2$ because $\mathbb R^2$ represnt $\mathbb C$. On the contrary, $N\ge1$ is arbitrary. – Denis Serre Jan 31 '11 at 21:11