David Corfield asked the following questions yesterday: Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\Bbb R^n$? What is its support?

I want to ask a more general question: what happens when $\|\cdot\|$ is the $p$-norm, for arbitrary $p\in [1, 2]$?

David's question, and Josh Shadlen's helpful answer, are here: Is the Fourier transform of $\exp(-\|x\|)$ non-negative?