New Bounds for the Traveling Salesman Constant

Let $X_1, X_2, \dots, X_n$ be independent and uniformly distributed random variables in the unit square $[0,1]^2$ and let $L(X_1, \dots, X_n)$ be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton $\&$ Hammersley proved the existence of a universal constant $\beta$ such that \lim_{n \rightarrow \infty}{n^{-1/2}L(X_1, \dots, X_n)} = \beta \qquad \mbox{almost surely.} The best bounds for $\beta$ are still the ones originally established by Beardwood, Halton $\&$ Hammersley $0.625 \leq \beta \leq 0.922$. We slightly improve both upper and lower bounds

Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts

Let $\Omega \subset \mathbb{R}^n$ be a convex. If $u: \Omega \rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincar\'{e} inequality \|u \|_{L^p} \leq c_p \mbox{diam}(\Omega) \| \nabla u \|_{L^p} with sharp constants $c_2 = 1/\pi$ (Payne \& Weinberger, 1960) and $c_1 = 1/2$ (Acosta \& Duran, 2005) independent of the dimension. The sharp constants $c_p$ for $1 < p < 2$ have recently been found by Ferone, Nitsch \& Trombetti (2012). The purpose of this short paper is to prove a much stronger inequality in the endpoint $L^1$: we combine results of Cianchi and Kannan, Lov\'{a}sz \& Simonovits to show that $$\left\|u\right\|_{L^{1}(\Omega)} \leq \frac{2}{\log{2}} M_{}(\Omega) \left\|\nabla u\right\|_{L^{1}(\Omega)}$$ where $M_{}(\Omega)$ is the average distance between a point in $\Omega$ and the center of gravity of $\Omega$. If $\Omega$ is a simplex, this yields an improvement by a factor of $\sim \sqrt{n}$ in $n$ dimensions. By interpolation, this implies that that for every convex $\Omega \subset \mathbb{R}^n$ and every $u:\Omega \rightarrow \mathbb{R}$ with mean 0 \left\|u\right\|_{L^{p}(\Omega)}\leq \left(\frac{2}{\log{2}} M_{}(\Omega) \right)^{\frac{1}{p}}\mbox{diam}(\Omega)^{1-\frac{1}{p}}\left\|\nabla u\right\|_{L^{p}(\Omega)}. Comment: New version with extension to L^p for p > 1, to be published in Archiv der Mathemati