For an isotropic convex body $K\subset {\mathbb{R}}^n$ we consider the isotropic constant $L_{K_N}$ of the symmetric random polytope $K_N$ generated by $N$ independent random points which are distributed according to the cone probability measure on the boundary of $K$. We show that with overwhelming probability $L_{K_N}\le C\sqrt{log(2N∕n)}$, where $C\in (0,\infty )$ is an absolute constant. If $K$ is unconditional we argue that even $L_{K_N}\le C$ with overwhelming probability and thereby verify the hyperplane conjecture for this model. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new ${\psi }_2$-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.

对于各向同性凸体 $ķ\subset {\mathbb{[R}}^{ñ}$ 我们考虑各向同性常数 ${\mathrm{大号}}_{{ķ}_{ñ}}$ 对称随机多态性 ${ķ}_{ñ}$ 由...产生 $ñ$ 独立的随机点，根据圆锥概率测度分布在 $ķ$。我们显示出压倒性的可能性${\mathrm{大号}}_{{ķ}_{ñ}}\le C\sqrt{日志（2ñ∕ñ）}$，在哪里 $C\in （0，\infty ）$是一个绝对常数。如果$ķ$ 是无条件的，我们认为，即使 ${\mathrm{大号}}_{{ķ}_{ñ}}\le C$具有压倒性的概率，从而验证了该模型的超平面猜想。证明分别基于亚指数或亚高斯随机变量之和的浓度不等式，在无条件情况下，基于新的${\psi }_2$-根据鲍勃科夫（Bobkov）和纳扎罗夫（Nazarov）的精神，针对锥度的线性泛函估计，这可能是与个人利益相关的。