Radon transform and Log-concavity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:56:10Z http://mathoverflow.net/feeds/question/53890 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53890/radon-transform-and-log-concavity Radon transform and Log-concavity Denis Serre 2011-01-31T14:37:52Z 2011-01-31T14:37:52Z <p>This question is related to (but different from) <a href="http://mathoverflow.net/questions/452" rel="nofollow">that of</a> Darsh Ranjan.</p> <p>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 <em>log-concave</em> if $\log g$ is concave. It is OK for me if one assumes that $f$ is integrable with compact support.</p> <p>The motivation comes from a <a href="http://www.umpa.ens-lyon.fr/~serre/DPF/Num5_2.pdf" rel="nofollow">paper</a> in collaboration with Th. Gallay, to appear in CPAM. With each $N\times N$ complex matrix $A$, we associate a <em>numerical measure</em>, which is a probability supported by the <em>numerical range</em> $W(A)\subset\mathbb C$. 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.</p>