AI 中文总结
研究高斯最大值的随机比较，通过结合高斯乘积不等式与自适应倾斜论证，解决了弱单纯形猜想，证明了单纯形平均宽度猜想中的不等式，并给出特定条件下可发送等概率消息最大数量的精确公式。
AI 中文摘要
我们证明了高斯最大值的随机比较。设\(R\)是一个\(m\times m\)的相关矩阵，满足\(R - \mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0\)，\(X\sim\mathcal{N}(0,R)\)，\(Z_1,\ldots,Z_m\)是独立标准高斯随机变量。则\(\max_{1\leq i\leq m}X_i \leq_{\mathrm{st}} \max_{1\leq i\leq m}Z_i\)。此比较解决了弱单纯形猜想，还证明了单纯形平均宽度猜想中的不等式，并给出了在码字能量约束下，确定性无反馈加性高斯白噪声码在规定能量和错误概率下可发送的等概率消息最大数量的精确公式。证明结合了对数凹函数的高斯乘积不等式和自适应倾斜论证。
英文摘要
Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. We prove $\max_i X_i\leq_{\mathrm{st}}\max_i Z_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint. The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\leq c\mathbf{1}\}>Φ(c)^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at https://github.com/abhmul/weak-simplex-conjecture-lean.
Comments42 pages
