双圆柱覆盖的斜率场扰动

双圆柱覆盖的斜率场扰动

Slope-field perturbations of the two-cylinder covering

OpenAI 数学仓库收录的研究手稿。展开查看英文摘要，或标记后让我进一步解释。

具体内容

稿件研究正四面体的双圆柱覆盖。作者声称可通过斜率场扰动得到有限覆盖，使其总垂直底面积严格小于最小投影面积的一半。这些覆盖对半面积问题及方向归一化半界猜想均给出否定回答。由仿射不变性，方向性结论对每个非退化四面体成立。该结果否定了圆柱覆盖中的半面积猜想。

• 扰动双圆柱覆盖得到总垂直底面积小于最小投影面积一半的有限覆盖。
• 否定半面积问题与方向归一化半界猜想。
• 由仿射不变性，结论对每个非退化四面体成立。

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英文原始摘要

The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.

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