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Register a new userAsymptotic Properties of Discrete Minimal $s,log^t$-Energy Constants and Configurations
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Combining the ideas of Riesz $s$-energy and $log$-energy, we introduce the so-called $s,log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,log^t$-energy constants and configurations of an infinite compact metric space of diameter less than $1$. In particular, we study certain continuity and differentiability properties of minimal $N$-point $s,log^t$-energy constants in the variable $s$ and we show that in the limits as $srightarrow infty$ and as $srightarrow s_0>0,$ minimal $N$-point $s,log^t$-energy configurations tend to an $N$-point best-packing configuration and a minimal $N$-point $s_0,log^t$-energy configuration, respectively. Furthermore, the optimality of $N$ distinct equally spaced points on circles in $mathbb{R}^2$ for some certain $s,log^t$ energy problems was proved.
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We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $Asubset mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $sin [p-2, p-1)$, we prove that the order of separation (as $Nto infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for `greedy $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.
Let $x_1,ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $mu $ on ${mathbb R}^n$, and consider the random polytope $$K_N:={rm conv}{ pm x_1,ldots ,pm x_N}.$$ We provide sharp estimates for the quermass{}integrals and other geometric parameters of $K_N$ in the range $cnls Nlsexp (n)$; these complement previous results from cite{DGT1} and cite{DGT} that were given for the range $cnls Nlsexp (sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(mu )$ of $mu $ for all $1ls qls n$.
Observations suggest that configurations of points on a sphere that are stable with respect to a Riesz potential distribute points uniformly over the sphere. Further, these stable configurations have a local structure that is largely hexagonal. Minimal configurations differ from stable configurations in the arrangement of defects within the hexagonal structure. This paper reports the asymptotic difference between the average energy of stable states and the lowest reported energies. We use this to infer the energy scale at which defects in the hexagonal structure are manifest. We report results for the Riesz potentials for s=0, s=1, s=2 and s=3. Additionally we compare existing theory for the asymptotic expansion in N of the minimal $N$-point energy with experimental results. We report a case of two distinct stable states that have the same Voronoi structure. Finally, we report the observed growth of the number of stable states as a function of N.
In this work we provide a way to introduce a probability measure on the space of minimal fillings of finite additive metric spaces as well as an algorithm for its computation. The values of probability, got from the analytical solution, coincide with the computer simulation for the computed cases. Also the built technique makes possible to find the asymptotic of the ratio for families of graph structures.
For $N$-point best-packing configurations $omega_N$ on a compact metric space $(A,rho)$, we obtain estimates for the mesh-separation ratio $gamma(omega_N,A)$, which is the quotient of the covering radius of $omega_N$ relative to $A$ and the minimum pairwise distance between points in $omega_N$. For best-packing configurations $omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $sto infty$, we prove that $gamma(omega_N,A)le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $omega_5^*$, that is the limit (as $sto infty$) of 5-point $s$-energy minimizing configurations. Moreover, $gamma(omega_5^*,S^2)=1$.
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