arXiv subjects
Publications and source records attributed to P. Dragnev.
In this article, we show that the minimal vectors of the extremal even unimodular lattices in $\mathbb{R}^{32}$ are $T$-avoiding universally optimal for suitable sets $T$. Moreover, they are minimal $T$-avoiding spherical designs and maximal $T$-avoiding codes for appropriate choices of $T$.
We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on $\mathbb{S}^{n-1}$ and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to $p$-frame energy.
We prove the universal optimality of four remarkable spherical 11-designs in 48 dimensions either among all antipodal codes, or all spherical 3-designs, whose inner-products avoid the set $T_1=(-1/3,-1/6) \cup (1/6,1/3)$. We also prove the universal optimality of these configurations among all codes whose distance-avoiding set is $T_2=(-1/2,-1/3) \cup (1/3,1/2)$.