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

I want to ask a more general question: what happens when ||.|| 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?