We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min-max-
min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^s$ with $s>0.$
Let $A$ be a compact $d$-rectifiable set embedded in Euclidean space $RR^p$, $dle p$. For a given continuous distribution $sigma(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our earlier results provided a method for generating $N$-po
int configurations on $A$ that have asymptotic distribution $sigma (x)$ as $Nto infty$; moreover such configurations are quasi-uniform in the sense that the ratio of the covering radius to the separation distance is bounded independent of $N$. The method is based upon minimizing the energy of $N$ particles constrained to $A$ interacting via a weighted power law potential $w(x,y)|x-y|^{-s}$, where $s>d$ is a fixed parameter and $w(x,y)=left(sigma(x)sigma(y)right)^{-({s}/{2d})}$. Here we show that one can generate points on $A$ with the above mentioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $r_N=C_N N^{-1/d}$ from each other, with $C_N$ being a positive sequence tending to infinity arbitrarily slowly. To do this we minimize the energy with respect to a varying truncated weight $v_N(x,y)=Phi(left|x-yright|/r_N)w(x,y)$, where $Phi:(0,infty)to [0,infty)$ is a bounded function with $Phi(t)=0$, $tgeq 1$, and $lim_{tto 0^+}Phi(t)=1$. This reduces, under appropriate assumptions, the complexity of generating $N$ point `low energy discretizations to order $N C_N^d$ computations.
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 p
airwise 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, $gamma(omega_5^*,S^2)=1$.
We prove a conjecture of Ambrus, Ball and Erd{e}lyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form sum_{k=1}^n f(d(z,z_k)), where $f:[0,pi]to [0,infty]$ is non-increasin
g and strictly convex and $d(z,w)$ denotes the geodesic distance between $z$ and $w$ on the circle.
In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+infty],$ the $N$-point $f$-best-packing constant $min{f(|x-y|), :, x,yin R^d}$, where the minimum is taken over p
oint sets of cardinality $N.$ We also show that $$ N^{1/d}Delta_d^{-1/d}-2le D_d(N)le N^{1/d}Delta_d^{-1/d}, quad Nge 2,$$ where $Delta_d$ is the maximal sphere packing density in $R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $Ntoinfty$.
We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz $s$-energy of $N$ points on the unit sphere in $mathbb{R}^{d+1}$, $dgeq 1$. The conjectures are based
on analytic continuation assumptions (with respect to $s$) for the coefficients in the asymptotic expansion (as $Nto infty$) of the optimal $s$-energy.
For a closed subset $K$ of a compact metric space $A$ possessing an $alpha$-regular measure $mu$ with $mu(K)>0$, we prove that whenever $s>alpha$, any sequence of weighted minimal Riesz $s$-energy configurations $omega_N={x_{i,N}^{(s)}}_{i=1}^N$ on $
K$ (for `nice weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as $N$ grows large. Furthermore, if $K$ is an $alpha$-rectifiable compact subset of Euclidean space ($alpha$ an integer) with positive and finite $alpha$-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as $Nto infty$) a prescribed positive continuous limit distribution with respect to $alpha$-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if $A$ is a compact $C^1$ manifold without boundary, then there exists a sequence of $N$-point best-packing configurations on $A$ whose mesh-separation ratios have limit superior (as $Nto infty$) at most 2.
We derive the complete asymptotic expansion in terms of powers of $N$ for the geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed curve $Gamma$ in ${mathbb R}^p$, $pgeq2$, as $N to infty$. For $f$ decreasing and convex, su
ch a point configuration minimizes the $f$-energy $sum_{j eq k}f(d(mathbf{x}_j, mathbf{x}_k))$, where $d$ is the geodesic distance (with respect to $Gamma$) between points on $Gamma$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s eq 0$) and the geodesic logarithmic potential $log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.
