论文标题
4-D高斯随机矢量最大猜想和3-D单纯宽度猜想
The 4-D Gaussian Random Vector Maximum Conjecture and the 3-D Simplex Mean Width Conjecture
论文作者
论文摘要
我们证明了四维高斯随机矢量最大猜想。该猜想断言，在所有中心的高斯随机矢量中，$ x =（x_1，x_2，x_3，x_4）$带有$ e [x_i^2] = 1 $，$ 1 $，$ 1 \ le i \ le 4 $，期望$矩阵等于$ - \ frac {1} {3} $。作为直接结果，我们解决了三维单纯形的平均宽度猜想。后者的猜想是凸几何形状中的一个长期开放问题，它断言，在所有刻在三维单元欧几里得球中的简单中，常规单纯形具有最大的平均宽度。
We prove the four-dimensional Gaussian random vector maximum conjecture. This conjecture asserts that among all centered Gaussian random vectors $X=(X_1,X_2,X_3,X_4)$ with $E[X_i^2]=1$, $1\le i\le 4$, the expectation $E[\max(X_1,X_2,X_3,X_4)]$ is maximal if and only if all off-diagonal elements of the covariance matrix equal $-\frac{1}{3}$. As a direct consequence, we resolve the three-dimensional simplex mean width conjecture. This latter conjecture is a long-standing open problem in convex geometry, which asserts that among all simplices inscribed into the three-dimensional unit Euclidean ball the regular simplex has the maximal mean width.
