We consider linear problems in the worst case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the values of the operator uniformly on a convex and balanced set by means of algorithms that use at most $n$ such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show in this paper, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest to us for two reasons. First, the homogeneity allows us to extend any error bound on the unit ball to the full input space. Second, homogeneous algorithms are better suited to tackle problems on cones, a scenario that is far less understood than the classical situation of balls. We illustrate our results by several examples.

翻译：我们考虑最坏情形下的线性问题。即给定一个线性算子与一组允许的线性测量，我们希望通过最多使用n次此类测量的算法，在凸平衡集上一致逼近该算子的值。已知一般线性算法无法实现最优逼近。然而，如本文所示，通过齐次算法总能获得最优逼近。这引起我们两方面的兴趣：其一，齐次性使得单位球上的任意误差界可扩展到整个输入空间；其二，齐次算法更适合处理锥上的问题——这一场景远不如经典的球情形那样被充分理解。我们通过若干算例阐明结论。