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：我们证明了所有美元- cones 都存在严格的 H\ “ oldererian 错误 ” 。 令人惊讶的是, Expenters 在许多方面与先前的假设有不同之处; 此外, 他们将$p$- cones 点亮为具有3个维度但并不在4个或4个以上的属性的奇特例子。 我们用我们的错误界限分析了美元- norm 正规化中最小的方位问题, 我们的计算结果使我们能够计算出相应的 KL Expent 值, 之前无法获取的值为$p$ 。 另一个应用程序是一个( 相对的) 简单证据, 证明大部分美元- ones 既不是自成一体的, 也不是同质的。 我们的错误界限是在面部残余功能框架内取得的, 我们通过为一般设定一个最佳标准来扩展它, 由此导致的错误约束必须非常严谨。