Complete sets need not be reduced in Minkowski spaces

TL;DR

This paper demonstrates that in Minkowski spaces of dimension three or higher, there exist complete sets that are not reduced, contrasting with Euclidean spaces where these classes coincide.

Contribution

It constructs explicit examples of complete sets that are not reduced in Minkowski spaces of dimension three or higher, showing the classes do not always coincide.

Findings

Complete sets need not be reduced in Minkowski spaces for n ≥ 3.

Constructed explicit examples of non-reduced complete sets.

Highlights difference between Euclidean and Minkowski geometries.

Abstract

It is well known that in $n$-dimensional Euclidean space ($n\geq 2$) the classes of (diametrically) complete sets and of bodies of constant width coincide. Due to this, they both form a proper subfamily of the class of reduced bodies. For $n$-dimensional Minkowski spaces, this coincidence is no longer true if $n\geq 3$. Thus, the question occurs whether for $n\geq 3$ any complete set is reduced. Answering this in the negative for $n\geq 3$, we construct $(2^{k}-1)$-dimensional ($k\geq 2$) complete sets which are not reduced.