AI 中文总结
研究正则交叉多胞体低维正交投影体积的上界，通过对绝对凸包边界三角剖分，比较相关行列式与高斯立体角，利用径向锥形成完全扇形得出结论：投影到\(k\)维子空间体积至多为\(2^k/k!\)，仅坐标子空间取等号。
AI 中文摘要
我们证明了关于正则交叉多胞体任意低维正交投影体积的猜想尖锐上界。更一般地，对于每个生成族\(v_1,\dots,v_n \in \mathbb{R}^k\)，我们证明\(\operatorname{vol}\nolimits_{k} \operatorname{conv} \{\pm v_1, \dots, \pm v_n\} \le \frac{2^k}{k!} \sqrt{\det\!\left(\sum_{i=1}^n v_i\otimes v_i\right)}\)。经过自然归一化，当非零向量构成正交基时等号成立。我们对绝对凸包的边界进行三角剖分，比较每个径向单纯形的行列式与其正锥的高斯立体角，然后利用径向锥形成一个完全扇形。结果，\(\crosp^n\)到任何\(k\)维子空间的投影体积至多为\(2^k/k!\)，仅坐标子空间取等号。
英文摘要
We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\dots,v_n \in \mathbb{R}^k, $ we prove \[ \operatorname{vol}\nolimits_{k} \operatorname{conv} \{\pm v_1, \dots, \pm v_n\} \le \frac{2^k}{k!} \sqrt{\det\!\left(\sum_{i=1}^n v_i\otimes v_i\right)}. \] After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every radial simplex with the Gaussian solid angle of its positive cone, and then use that the radial cones form a complete fan. As a consequence, the volume of the projection of $\crosp^n$ onto any $k$-dimensional subspace is at most $2^k/k!$, with equality only for coordinate subspaces.
