Error bounds play a central role in the study of conic optimization problems, including the analysis of convergence rates for numerous algorithms. Curiously, those error bounds are often H\"olderian with exponent 1/2. In this paper, we try to explain the prevalence of the 1/2 exponent by investigating generic properties of error bounds for conic feasibility problems where the underlying cone is a perspective cone constructed from a nonnegative Legendre function on $\mathbb{R}$. Our analysis relies on the facial reduction technique and the computation of one-step facial residual functions (1-FRFs). Specifically, under appropriate assumptions on the Legendre function, we show that 1-FRFs can be taken to be H\"olderian of exponent 1/2 almost everywhere with respect to the two-dimensional Hausdorff measure. This enables us to further establish that having a uniform H\"olderian error bound with exponent 1/2 is a generic property for a class of feasibility problems involving these cones.

翻译：误差界在锥优化问题的研究中扮演着核心角色，包括对众多算法收敛速率的分析。有趣的是，这些误差界通常是指数为1/2的赫尔德型误差界。本文试图通过研究一类锥可行性问题误差界的通用性质来解释指数1/2的普遍性，其中所涉及的锥是由$\mathbb{R}$上的非负勒让德函数构造的透视锥。我们的分析依赖于面约化技术和一步面残差函数（1-FRFs）的计算。具体而言，在对勒让德函数施加适当假设的条件下，我们证明1-FRFs几乎处处（相对于二维豪斯多夫测度）可以取为指数为1/2的赫尔德型函数。这使我们能够进一步证明，对于涉及这类锥的一类可行性问题而言，具有指数为1/2的一致赫尔德型误差界是一个通用性质。