For any norm on $\mathbb{R}^d$ with countably many extreme points, we prove that there is a set $E \subset \mathbb{R}^d$ of Hausdorff dimension $d$ whose distance set with respect to this norm has zero linear measure. This was previously known only for norms associated to certain finite polygons in $\mathbb{R}^2$. Similar examples exist for norms that are very well approximated by polyhedral norms, including some examples where the unit ball is strictly convex and has $C^1$ boundary.

中文翻译：

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Falconer 的 $(K, d)$ 距离集猜想对于 $\mathbb R^d$ 中的严格凸集 $K$ 可能失败

对于 $\mathbb{R}^d$ 上具有可数多个极值点的任何范数，我们证明存在 Hausdorff 维 $d$ 的集合 $E \subset \mathbb{R}^d$，其距离设置为该范数的线性度量为零。这之前仅针对与 $\mathbb{R}^2$ 中某些有限多边形相关联的范数已知。类似的例子存在于被多面体范数很好地近似的范数中，包括一些单位球是严格凸的并且具有 $C^1$ 边界的例子。