Immobilization of convex bodies in $R^n$

TL;DR

This paper generalizes the concept of immobilizing convex bodies in n-dimensional space, providing conditions for immobilization of simplices and revealing qualitative differences as dimension increases.

Contribution

It extends immobilization theory to arbitrary dimensions and characterizes conditions for n-simplex immobilization with geometric insights.

Findings

Necessary and sufficient conditions for n-simplex immobilization

Qualitative differences in immobilization sets as dimension increases

Geometric description of immobilization conditions

Abstract

We extend to arbitrary finite $n$ the notion of immobilization of a convex body $O$ in $R^n$ by a finite set of points $P$ in the boundary of $O$. Because of its importance for this problem, necessary and sufficient conditions are found for the immobilization of an $n$-simplex. A fairly complete geometric description of these conditions is given: as $n$ increases from $n = 2$, some qualitative difference in the nature of the sets $P$ emerges.