Fixing and almost fixing a planar convex body

TL;DR

This paper explores conditions under which points on the boundary of a planar convex body can fix or almost fix the body, providing first-order criteria and comparing different definitions of these concepts.

Contribution

It introduces and compares various definitions of fixing and almost fixing convex bodies and derives first-order conditions for these properties.

Findings

First-order conditions for fixing points on convex bodies

Comparison of different fixing and almost fixing definitions

Theoretical criteria for boundary point configurations

Abstract

A set of points a 1 ,. .. , a n fixes a planar convex body K if the points are on bdK, the boundary of K, and if any small move of K brings some point of the set in intK, the interior of K. The points a 1 ,. .. , a n $\in$ bdK almost fix K if, for any neighbourhoods V i of a i (i = 1,. .. , n), there are pairs of points a i , a i $\in$ V i $\cap$ bdK such that a 1 , a 1 ,. .. , a n fix K. This note compares several definitions of these notions and gives first order conditions for a 1 ,. .. , a n $\in$ bdK to fix, and to almost fix, K.