照亮 ${\mathbb R}^3$ 和 ${\mathbb R}^4$ 中的 1-无条件凸体，以及更高维度中的某些情形
Illuminating 1-unconditional convex bodies in ${\mathbb R}^3$ and ${\mathbb R}^4$, and certain cases in higher dimensions
AI总结：
本文解决了三维和四维空间中所有1-无条件凸体的照明猜想，并推广到具有特定坐标投影的更高维情形，确认了立方体类子集的等式情形，方法基于组合技术和深照明概念。
摘要：
我们解决了 ${\mathbb R}^3$ 和 ${\mathbb R}^4$ 中所有 1-无条件凸体的 Hadwiger-Boltyanski 照明猜想。此外，我们解决了那些至少有一个坐标超平面投影等于外接矩形盒相应投影的更高维 1-无条件凸体的猜想。最后，我们确认了在 1-无条件凸体子类中照明猜想的猜想等式情形，这些凸体与立方体 $[-1,1]^n$ 一样，在坐标子空间上没有极点。我们的方法是组合性的，我们使用的照明集主要由标准基向量的小扰动组成。特别是，我们基于 [Sun-Vritsiou, \"On the illumination of 1-symmetric convex bodies\", 预印本 arXiv:2407.10314] 中的思想和构造，主要基于其中引入的 '深照明' 概念。
英文摘要：
We settle the Hadwiger-Boltyanski Illumination Conjecture for all 1-unconditional convex bodies in ${\mathbb R}^3$ and in ${\mathbb R}^4$. Moreover, we settle the conjecture for those higher-dimensional 1-unconditional convex bodies which have at least one coordinate hyperplane projection equal to the corresponding projection of the circumscribing rectangular box. Finally, we confirm the conjectured equality cases of the Illumination Conjecture within the subclass of 1-unconditional bodies which, just like the cube $[-1,1]^n$, have no extreme points on coordinate subspaces. Our methods are combinatorial, and the illuminating sets that we use consist primarily of small perturbations of the standard basis vectors. In particular, we build on ideas and constructions from [Sun-Vritsiou, "On the illumination of 1-symmetric convex bodies", preprint available at arXiv:2407.10314], and mainly on the notion of 'deep illumination' introduced there.
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