On geometric properties of sets of positive reach in ${\bf E}^d$

Details On geometric properties of sets of positive reach in ${\bf E}^d$ On geometric properties of sets of positive reach in ${\bf E}^d$

2007-03-21

arxiv.org/abs/math/0703634v1

Mathematics

Metric Geometry

10 pages, 2 figures

Scientific paper

Some geometric facts concerning sets with positive reach in the Euclidean $d$-dimensional space ${\bf E}^d$ are proved. For $x_1$ and $x_2$ in ${\bf E}^d$ and $R>0$ let us denote by ${\mathfrak H}(x_1,x_2,R)$ the intersection of all closed balls of radius $R$ containing $x_1$ and $x_2$. For a compact subset $K$ of ${bf E}^d$ we prove that ${\rm reach}(K)\ge R$ if and only if for every $x_1,x_2\in K$ such that $\Vert x_1-x_2\Vert< 2R$, ${\mathfrak H}(x_1,x_2,R)\cap K$ is connected. A corollary is that if ${\rm reach}(K)\ge R>0$ and $D$ is a closed ball of radius less than or equal to $R$ (intersecting $K$) then ${\rm reach}(K\cap D)\ge R$. We also give a necessary and sufficient condition such that $A\subset{\bf E}^d$ admits a minimal cover (with respect to inclusion) of ${\rm reach}\ge R$.

Affiliated with

Also associated with

No associations

Rating

On geometric properties of sets of positive reach in ${\bf E}^d$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On geometric properties of sets of positive reach in ${\bf E}^d$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On geometric properties of sets of positive reach in ${\bf E}^d$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-382737