Fixing and almost fixing a planar convex body

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.