A convex polyhedron without Rupert's property

TL;DR

This paper constructs a convex polyhedron that does not have Rupert's property, disproving a 2017 conjecture, and also identifies a polyhedron that is Rupert but not locally Rupert, advancing understanding of geometric properties.

Contribution

It provides the first explicit example of a convex polyhedron without Rupert's property and clarifies distinctions between Rupert and locally Rupert polyhedra.

Findings

Constructed a convex polyhedron without Rupert's property

Disproved the 2017 conjecture regarding Rupert's property

Identified a polyhedron that is Rupert but not locally Rupert

Abstract

A three-dimensional convex body is said to have Rupert's property if its copy can be passed through a straight hole inside that body. In this work we construct a polyhedron which is provably not Rupert, thus we disprove a conjecture from 2017. We also find a polyhedron that is Rupert but not locally Rupert.