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We prove tight H\"olderian error bounds for all $p$-cones. Surprisingly, theexponents differ in several ways from those that have been previouslyconjectured; moreover, they illuminate $p$-cones as a curious example of aclass of objects that possess properties in 3 dimensions that they do not in 4or more. Using our error bounds, we analyse least squares problems with$p$-norm regularization, where our results enable us to compute thecorresponding KL exponents for previously inaccessible values of $p$. Anotherapplication is a (relatively) simple proof that most $p$-cones are neitherself-dual nor homogeneous. Our error bounds are obtained under the framework offacial residual functions, and we expand it by establishing for general conesan optimality criterion under which the resulting error bound must be tight.

Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond