I was just surfing on your very handy Big Picture page. Around the Section 3, "CS Measurements", I saw that you have listed the zoology of the different *RIP.
If you want one more, in the paper that I co-wrote with David Hammond and M. Jalal Fadili about Dequantizing CS, we used the RIP_p to characterize the performance of the BPDQ decoder. RIP_p is there just the "sandwiching" of the Lp norm (with p>=2) of the random projection of sparse signals x, i.e. Phi x, between their L2 norm multiplied by the common (1+- delta)^1/2 factors, so that RIP_2 is the common RIP.
I am going to add it in the Big Picture. As I have said before, I look forward to having more of this type of interaction, so that the Big Picture really becomes a living document on Compressive Sensing. According to the stats, that page has an average of 100 visitors per day!
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I noticed also that Rick Chartrand and Valentina Staneva, in
"Restricted isometry properties and nonconvex compressive sensing", Inverse Problems, vol. 24, no. 035020, pp. 1--14, 2008
introduced also a RIP_p definition, with an existence proof for Gaussian matrices, but .... for 0 < p <=1
In conclusion, the RIP_p makes sense for p in R^* now ;-)
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