Extremal slabs in the cube and the Laplace transform.
In this very interesting paper the authors study the volume of symmetric slabs in the unit cube. The work is motivated by the following question. Let \(Q_n=\left[-\frac{1}{2},\frac{1}{2}\right]^n\) be the unit cube in \(R^n\), and \(t\) positive. What is the minimal value of the volume of \(Q_n\cap\{x;\;| \langle x,u\rangle| \leq t\}\) over unit vectors \(u\in S^{n-1}\)? This problem was posed by Milman, and he conjectured that the canonical slab (section parallel to a face) is minimal among all symmetric slabs with fixed width \(t\), for \(t\) smaller than some absolute constant, not depending on the dimension. The authors, equivalently, look for the minimal volume of the intersection of the cube with origin-symmetric slabs \(S=\{x;\;| \langle x,u\rangle| \leq w/2\}\) of fixed width \(w\). It turns out that, if \(w\leq\frac{3}{4}\), then \(| Q_n\cap S| \geq | Q_n\cap S_0| =w\), where \(S_0=\{x;\;| x_1| \leq w/2\}\) is the canonical slab of width \(w\). This important result shows that Milman's conjecture holds true for \(t\) smaller than \(\frac{3}{4}\). For large width, the authors prove that the slab orthogonal to the main diagonal becomes asymptotically minimal as the dimension goes to infinity. A crucial role in the paper is played by the concavity or convexity of the function \(t\mapsto\log L_{\mu}(\sqrt{t})\) for some measure \(\mu\) on \(R\), where \(L_{\mu}\) is the Laplace transform of \(\mu\). Different classes of measure \(\mu\) having this property are investigated, and extensions of the results to other situations are also explored.
scientific article; zbMATH DE number 1895792
scientific article