发表机构
Stevens Institute of Technology（史蒂文斯理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定各维度下立方-顶点体的最小体积乘积，通过将高度集中在一对相对角顶点处达到最小值，结合Lean 4完成验证，还得到了无乘积条件的相关界。
AI 中文摘要
我们确定了在固定总外高度时，通过向立方体添加面顶点得到的中心对称多面体的最小体积乘积。高度和切向坐标受限于极帽分母的乘积条件。在所有维度至少为2的情况下，该最小值有显式公式，且通过将高度集中在一对相对的角顶点处达到。对于维度至少为3的正高度，所有极小化阵列均具有该结构。该乘积条件严格扩展了逐坐标二次约束。我们还得到了无该条件的两个界：指定高度范围下的四维Mahler界，以及衡量两个顶点共同生成体积的修正项，后者证明了存在超出前述充分条件的连续族。证明结合了已知的帽体积、严格的总高度估计和双棱锥构造，Lean 4验证了实际体积最小值、必要的等式结构及两个扩展。
英文摘要
We determine the minimum volume product of centrally symmetric polytopes obtained by adding facet apices to a cube while fixing their total outward height. Heights and tangential coordinates vary subject to a product condition on the polar-cap denominators. In every dimension at least two, the minimum has an explicit formula, attained by concentrating the height at one opposite pair of corner apices. For positive height in dimensions at least three, every minimizing array has this structure. The product condition strictly extends coordinatewise quadratic constraints. We also obtain two bounds without that condition: a four-dimensional Mahler bound for a specified height range, and a correction measuring volume generated jointly by two apices. The latter certifies a continuous family outside the preceding sufficient conditions. The proof combines known cap volumes with a sharp aggregate height estimate and a double-pyramid construction. Lean~4 verifies the actual-volume minimum, the necessary equality structure and both extensions.
Comments5 pages, 1 figure
