It is shown that for any sufficiently regular even Minkowski valuation Φ which is homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are the only solutions to the fixed-point problem ${\mathrm{\Phi }}^{2}K=\alpha K$. This significantly generalizes results by Ivaki for projection bodies and suggests, via the Lutwak–Schneider class reduction technique, a new approach to Petty's conjectured projection inequality.

中文翻译：

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Minkowski 估值的固定点

结果表明，对于任何均匀且交织刚性运动的足够规则的偶数 Minkowski 估值 Φ，都存在单位球的邻域，其中球是不动点问题的唯一解 ${\mathrm{\Phi }}^2钾=\alpha 钾$. 这极大地概括了 Ivaki 对投影体的结果，并通过 Lutwak-Schneider 类缩减技术提出了一种解决 Petty 推测投影不等式的新方法。