Summary: Given a connected graph \(G=(V,E)\) with positive edge weights, define the distance \(d_ G(u,v)\) between vertices \(u\) and \(v\) to be the length of a shortest path from \(u\) to \(v\) in \(G\). A spanning subgraph \(G'\) of \(G\) is said to be a \(t\)-spanner for \(G\) if, for every pair of vertices \(u\) and \(v\), \(d_{G'}(u,v)\leq t\cdot d_ G(u,v)\). Consider a complete graph \(G\) whose vertex set is a set of \(n\) points in \({\mathfrak R}^ k\) and whose edge weights are given by the \(L_ p\) distance between respective points. Given input parameter \(\epsilon\), \(0<\epsilon\leq 1\), we show how to construct a \((1+\epsilon)\)-spanner for \(G\) containing \(O({n \over \epsilon^{2k}})\) edges in \(O(n\log n+{n \over \epsilon^{2k}})\) time. We apply this spanner to the construction of approximate minimum spanning trees.