Rendezvous numbers of metric spaces - a potential theoretic approach

Potential theoretic approach to rendezvous numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x1,...,xn} in X, there exists some point x in X with the average of the distances d(x,xj) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named "the magic number"…

Rendezvous numbers in normed spaces

In previous papers, we used abstract potential theory, as developed by Fuglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We introduced energies, Chebyshev constants as two variable set functions, and the modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces. Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in a…

环面与球面上数论点集在Wasserstein度量和L²偏差下的等分布

Equidistribution of number theoretic point sets in the torus and the sphere in Wasserstein metric and $L^2$ discrepancy

该研究确定了环面与球面上Kronecker序列、Fibonacci格点等算术点集在Wasserstein度量和L²偏差下的等分布速率，证明了两类度量间的严格不等式及对应最优性关联。

Aspects of area formulas by way of Luzin, Radó, and Reichelderfer on metric measure spaces

We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincaré inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas.