Acute Sets In Euclidean Spaces

Abstract

A finite set H in R d is called an acute set if any angle determined by three points of H is acute. We examine the maximal cardinality αðdÞ of a d-dimensional acute set. The exact value of αðdÞ is known only for d ≤ 3. For each d ≥ 4 we improve on the best known lower bound for αðdÞ. We present different approaches. On one hand, we give a probabilistic proof that αðdÞ > c · 1:2 d . (This improves a random construction given by Erdős and Füredi.) On the other hand, we give an almost exponential constructive example which outdoes the random construction in low dimension (d ≤ 250). Both approaches use the small dimensional examples that we found partly by hand (d ¼ 4, 5) and partly by computer (6 ≤ d ≤ 10). We also investigate the following variant of the above problem: what is the maximal size κðdÞ of a d-dimensional cubic acute set (that is, an acute set contained in the vertex set of a d-dimensional hypercube)? We give an almost exponential constructive lower bound, and we improve on the best known upper bound.

Explicit Constructions of Centrally Symmetric $$k$$ -Neighborly Polytopes and Large Strictly Antipodal Sets

“…(It is worth remarking that Erdős and Füredi established existence of an acute set in R d that has an exponential size in d. As every acute set has the property that all of its pairs of vertices are strictly antipodal, their result implied an exponential lower bound on A ′ (d). A significant improvement of the Erdős-Füredi bound on the maximum size of an acute set in R d was recently found by Harangi [10]. )…”

Acute Sets of Exponentially Optimal Size

“…(See http://dxdy.ru/post1222167.html#p1222167 and http://dxdy.ru/post1231694.html#p1231694 for these examples.) Previously, the best known lower bounds were 8 in dimension 4 and 13 in dimension 5, see [6,7]. His idea was to start from the vertices of a (d − 1)-dimensional hypercube and slightly modify the coordinates.…”

The number of double-normals in space

“…It is also clear that ( ) ≤ ′ ( ) ≤ ′′ ( ). From the result of Danzer and Grünbaum we know that ( ) = ⌈log 2 ⌉, while from the result of V. Harangi [4], who improved the bound due to Erdős and Füredi to ′ ( ) ≥ 1.2 for large , we have ′ ( ) ≤ log 1.2 for large .…”

“…Although Theorem 1 gives 𝑘(3) = 𝑘 ′ (3) = 2, it does not allow to decide whether 𝑘 (4) or 𝑘 ′ (4) equals 2 or 3. In this paper we improve both the upper and the lower bounds from Theorem 1, which gives, in particular, 𝑘(4) = 𝑘 ′ (4) = 2.…”