## 2 Preliminaries  
To define the metric average of two sets, we first introduce some notation. The _Euclidean distance_ from a point \(p\) to a set \(A\subset R^{n}\) is  
\[\text{dist}\left(p,A\right)=\inf\left\{\left|p-q\right|:q\in A\right\},\]with \(|\cdot|\) the Euclidean norm. A point \(q\) is called a _projection point_ of a point \(p\) on a set \(A\), if \(q\) belongs to the _closure_ of \(A\) and \(|p-q|=dist\left(p,A\right)\). Generally the projection point is not unique, and the set of all projection points of \(p\) on \(A\) is denoted by \(\Pi_{A}\left(p\right)\). For two sets \(A,B\) and \(t\in\left[0,1\right]\), we introduce the asymmetric operation \(M_{t}(A,B)\), which is the set of all \(t\)-weighted averages between any point in \(A\) and all its projection points on \(B\), namely  
\[M_{t}(A,B)=\left\{tp+\left(1-t\right)q:p\in A,q\in\Pi_{B}\left(p\right)\right\}. \tag{2.1}\]  
Finally the \(t\)-weighted _metric average_ of two compact sets \(A,B\) is defined as,  
\[A\oplus_{t}B=M_{t}\left(A,B\right)\bigcup M_{1-t}\left(B,A\right). \tag{2.2}\]  
It is easy to verify that \(A\oplus_{t}A=A\), \(A\oplus_{1}B=A\), and \(A\oplus_{0}B=B\). An example of the metric average of two sets in 1D is given in Figure 1.  
Since for \(p\in A\bigcap B\), the projection point of \(p\) on \(A\), and also the projection point of \(p\) on \(B\) is the point \(p\) itself, we have  
\[A\oplus_{t}B=\left(A\bigcap B\right)\bigcup M_{t}(A\setminus B,B)\bigcup M_{ 1-t}(B\setminus A,A), \tag{2.3}\]  
where \(\setminus\) denotes the usual set difference. Now since for points outside a set, the projection on the set is equivalent to the projection on the boundary of the set, relation (2.3) can be further rewritten as,  
\[A\oplus_{t}B=\left(A\bigcap B\right)\bigcup M_{t}\left(A\setminus B,\partial B \right)\bigcup M_{1-t}\left(B\setminus A,\partial A\right), \tag{2.4}\]  
where \(\partial\) denotes the boundary of a set. Next we briefly recall the definitions of the geometric objects used in this work. A collection of points \(p=\left(x,y\right)\in\mathbb{R}^{2}\) satisfying  
\[ax^{2}+by^{2}+cxy+dx+ey+f=0,\]  
with \(a,b,c,d,e,f\in\mathbb{R}\), is called a _planar conic curve_. We refer to segments of conic curves as _conic segments_. A _simple polygon_ is a region of the plane bounded by a single closed and not self-intersecting chain of linear segments, termed _edges_. A _simple polygon with holes_ is a simple polygon that contains holes,  
**Figure 1.** The metric average of two non-convex sets in 1D, where \(A=\left[0,1\right]\bigcup\left[2,3\right]\), \(B=\left[0.5,1.3\right]\bigcup\left[1.7,2.5\right]\), \(A\oplus_{1/2}B=\left[0.25,1.15\right]\bigcup\left[1.85,2.75\right]\). which are simple polygons. If we allow the edges to be conic segments, we get a _simple conic polygon with holes_. A set with piecewise linear boundaries is a collection of pairwise disjoint simple polygons with holes. A collection of simple conic polygons with holes can be treated as a _planar arrangement_ (_c.f._[9]). Given a set \(C\) of bounded planar curves, the arrangement \(A(C)\) is the subdivision of the plane induced by the curves in \(C\) into zero-dimensional, one-dimensional and two-dimensional cells, called _vertices_, _edges_ and _faces_ respectively. The _overlay_ of two arrangements \(A(C_{1}),A(C_{2})\) is the arrangement \(A(C_{1}\bigcup C_{2})\) produced by the edges from \(C_{1}\) and \(C_{2}\). Given a collection \(\Sigma\) of geometric objects in \(\mathbb{R}^{2}\), called _Voronoi sites_, the _Voronoi diagram_ (see e.g. [8, 11]) of \(\Sigma\) is the subdivision of the plane into regions (called _Voronoi faces_), each region being associated with a site \(s_{i}\in\Sigma\), and containing all points of the plane for which \(s_{i}\) is closest (with respect to the Euclidean distance) among all the sites in \(\Sigma\). A _Voronoi bisector_ is a set of points that are equidistant from two Voronoi sites. The Voronoi diagram induced by a collection of linear segments in \(\mathbb{R}^{2}\) is termed _segment Voronoi diagram_.