各向同性凸体的次高斯体的几何
Geometry of the subgaussian body of an isotropic convex body
AI总结：
该研究针对各向同性凸体$K$的$\u03a8_2(K)$体，证明其与$L_2$质心体$Z_2(K)$有界体积比，得到其平均宽度等精确估计，还构造了次高斯常数受控的标准正交基。
AI中文摘要：
对于中心凸体$K\ubded{mathbb{R}^n}$，令$\u03a8_2(K)$表示对称凸体，其支撑函数由$K$上线性泛函的$\u03c8_2$-范数给出。Letwin和Mikulincer近期解决了关于次高斯方向存在性的Milman问题，这自然推动了对该体几何的研究。我们证明$\u03a8_2(K)$相对于$L_2$质心体$Z_2(K)$具有有界体积比。在各向同性情形下，我们还得到其平均宽度及其正交投影体积半径的精确估计，并推导次高斯标准正交基存在性的相关结论；特别地，我们为每个各向同性凸体构造了次高斯常数受定量控制的标准正交基。
英文摘要：
For a centered convex body $K\subset\mathbb{R}^n$, let $Ψ_2(K)$ denote the symmetric convex body whose support function is given by the $ψ_2$-norms of linear functionals on $K$. The recent solution of Milman's problem on the existence of subgaussian directions by Letwin and Mikulincer naturally motivates the study of the geometry of this body. We prove that $Ψ_2(K)$ has bounded volume ratio with respect to the $L_2$-centroid body $Z_2(K)$. In the isotropic case, we also obtain sharp estimates for its mean width and the volume radii of its orthogonal projections, and derive consequences for the existence of subgaussian orthonormal bases. In particular, we construct orthonormal bases with quantitatively controlled subgaussian constants for every isotropic convex body.
