AI 中文总结
本文通过支撑函数的负矩估计和球冠论证，研究各向同性凸体的随机投影内半径等几何量，得到多项式偏差估计并证明立方体在无条件类中的极值性，同时给出混合旋转-投影定理和体积剖面的尖锐结果。
AI 中文摘要
我们研究各向同性凸体的支撑函数的下尾。一个端点负矩估计结合球冠论证，给出了任意维度下随机投影的内半径的多项式偏差估计；在无条件类中，这产生了中位内半径的最优阶，并表明立方体是极值情形。同样的论证给出了独立旋转的凸包的估计，并在额外投影后，得到一个混合旋转-投影定理。我们还证明了旋转极体的闵可夫斯基平均及其随机投影的加权下界和上界估计，以及一个依赖于内半径的全局平均到欧氏球的几何距离的估计。一族各向同性乘积圆柱表明，后者在内半径的整个可能范围内，直到绝对常数都是精确的。对于立方体，一个显式的负矩计算给出了定量的投影$(m,k,n)$-profile，并在全维情形下恢复了有界几何距离的经典阶$n/\n(en)$。最后，我们在无条件各向同性类中确定了独立旋转交集的尖锐体积剖面，并表明立方体是极值情形。
英文摘要
We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected $(m,k,n)$-profile and, in full dimension, recovers the classical order $n/\ln(en)$ for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.
