Abstract. A spherical [Formula: see text]- design is a finite subset [Formula: see text] of the unit sphere such that every polynomial of degree at most [Formula: see text] has the same average over [Formula: see text] as it does over the entire sphere. Determining the minimum possible size of spherical designs, especially in a fixed dimension as [Formula: see text], has been an important research topic for several decades. This paper presents results on the complementary asymptotic regime, where [Formula: see text] is fixed and the dimension tends to infinity. The main results in this paper are (1) a construction of smaller spherical designs via an explicit connection to Gaussian designs and (2) the exact order of magnitude of minimal-size signed [Formula: see text]-designs, which is significantly smaller than predicted by a typical degrees-of-freedom heuristic. We also establish a method to “project” spherical designs between dimensions, prove a variety of results on approximate designs, and construct new [Formula: see text]-wise independent subsets of [Formula: see text] which may be of independent interest. To achieve these results, we combine techniques from algebra, geometry, probability, representation theory, and optimization.

A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a long-standing conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of “containing $k$ colinear integer points.”