We prove tight H\"olderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with $p$-norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of $p$. Another application is a (relatively) simple proof that most $p$-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.

翻译：我们证明了所有 $p$-锥的严格Hölder误差界。令人惊讶的是，这些指数在多个方面与先前推测的结果不同；此外，它们揭示了 $p$-锥是一类有趣的对象，其在三维空间中具有的性质在四维或更高维度中并不存在。利用我们的误差界，我们分析了带有 $p$-范数正则化的最小二乘问题，我们的结果使我们能够针对先前无法访问的 $p$ 值计算相应的KL指数。另一个应用是（相对）简单地证明了大多数 $p$-锥既不是自对偶的也不是齐次的。我们的误差界是在面残差函数的框架下获得的，并且我们通过为一般锥建立最优性准则来扩展该框架，在该准则下所得误差界必须是紧的。