发表机构
Charles University, Faculty of Mathematics and Physics, Mathematical Institute of Charles University（查理大学数学与物理学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究凸体混合体积的高阶Godbersen猜想，在低维等特殊情形下证明猜想，推广加权不等式，并引入高阶非平衡并，提出插值新猜想。
AI 中文摘要
我们考虑凸体混合体积的Godbersen猜想的两个高阶推广，这些猜想最早由Schneider于2000年提出。首先，我们在若干特殊情形下，特别是在低维情形下，建立了这些猜想。其次，我们证明了猜想的某些推论成立。更确切地说，我们定义了非平衡差体（unbalanced difference body）的高阶版本，并证明了相关的加权不等式，推广了Artstein-Avidan和Putterman的先前结果。最后，我们引入了由Artstein-Avidan、Einhorn、Florentin和Ostrover所考虑的凸体非平衡并（unbalanced joins）的高阶类比，证明了相应的体积界，并提出了在高阶Godbersen猜想与Fáry和Rédei的一个猜想之间进行插值的猜想。
英文摘要
We consider two higher-order generalizations of the Godbersen conjecture for mixed volumes of convex bodies, which were first proposed by Schneider in 2000. First, we establish the conjectures in several special cases, in particular in low dimensions. Second, we prove that certain consequences of the conjectures hold. More precisely, we define a higher-order version of the unbalanced difference body and prove related weighted inequalities, generalizing previous results of Artstein-Avidan and Putterman. Finally, we introduce higher-order analogs of unbalanced joins of convex bodies considered by Artstein-Avidan, Einhorn, Florentin, and Ostrover, prove the corresponding volume bounds, and propose conjectures that interpolate between the higher-order Godbersen conjectures and a conjecture due to Fáry and Rédei.
Comments31 pages, minor changes
