Acute Sets of Exponentially Optimal Size

Abstract

We present a simple construction of an acute set of size 2 d−1 + 1 in R d for any dimension d. That is, we explicitly give 2 d−1 + 1 points in the d-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than 2 d . Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order ϕ d where ϕ = (1 + √ 5)/2 ≈ 1.618 is the golden ratio.2010 Mathematics Subject Classification. 51M04, 51M15.

From Acute Sets to Centrally Symmetric 2-Neighborly Polytopes

“…We start with the set S 0 described below; we then perturb the points of S 0 to obtain an almost acute set. As our construction/proof is a simple modification of that in [8], we only sketch the main ideas leaving out some of the details.…”

“…The author is grateful to a participant of the mathematical forum dxdy.ru, who wished to remain anonymous, for bringing reference [8] to our attention.…”

“…Although the size of the largest acute set in R d remains a mystery, in a very recent breakthrough paper [8], Gerencsér and Harangi constructed an acute set in R d of size 2 d−1 + 1. The previous record size was F d+2 = Θ ( 1+ √ 5…”

“…To prove Lemma 3 we modify the Gerencsér-Harangi construction: as in [8], we start with the vertex set of the…”

Exponentially sized pointsets with angles less than 61 degrees

“…As it contains a lot of angles equal to π/2, the following question becomes interesting: what is the maximum number of points in R d with all angles strictly less than π/2? Very recently, in 2017, Gerenscér and Harangi [4] made a breakthrough by providing a surprisingly simple construction of 2 d−1 + 1 points. Regarding an upper bound, currently it is not known even whether sets of 2 d − 1 points are possible.…”