We study the infimal value of the Hausdorff dimension of spaces that are Hölder equivalent to a given metric space; we call this bi‐Hölder‐invariant ‘Hölder dimension’. This definition and some of our methods are analogous to those used in the study of conformal dimension. We prove that Hölder dimension is bounded above by capacity dimension for compact, doubling metric spaces. As a corollary, we obtain that Hölder dimension is equal to topological dimension for compact, locally self‐similar metric spaces. In the process, we show that any compact, doubling metric space can be mapped into Hilbert space so that the map is a bi‐Hölder homeomorphism onto its image and the Hausdorff dimension of the image is arbitrarily close to the original space's capacity dimension. We provide examples to illustrate the sharpness of our results. For instance, one example shows Hölder dimension can be strictly greater than topological dimension for non‐self‐similar spaces, and another shows the Hölder dimension need not be attained.

中文翻译：

---

在Hölder等效条件下最小化Hausdorff尺寸

我们研究了等于给定度量空间的Hölder空间的Hausdorff维数的极小值；我们称其为双霍德尔不变的“霍德尔维数”。这个定义和我们的一些方法类似于保形维研究中使用的定义。我们证明，对于紧凑的，加倍的度量空间，Hölder维度受容量维度的限制。作为推论，我们得出Hölder维等于紧凑的，局部自相似度量空间的拓扑维。在此过程中，我们证明了可以将任何紧凑的，加倍的度量空间映射到希尔伯特空间，从而使映射成为图像上的双荷尔德同胚，并且图像的Hausdorff维数任意接近原始空间的容量维。我们提供了一些例子来说明结果的清晰度。例如，