On graphs in Euclidean spaces with minimal total curvature (Q1974068)

This paper deals with flat maps, that is, polygonal maps of finite graphs \(G\) into the Euclidean space \(E^n\), whose total curvature is minimal for \(G\). The author computes the curvature dimension \(\dim (K_m)\), defined by \textit{K. Taniyama} [Differ. Geom. Appl. 8, 135-155 (1998; Zbl 0924.53004)], and the vertex dimension \(\text{Verdim} (K_m)\), defined by the author in [Differ. Geom. Appl. 8, 163-170 (1998; Zbl 0921.05030)], for a complete graph \(K_m\) to be \(\dim(K_{2i}) =\dim(K_{2i-1}) =1+ {1\over 2}i(i-1)\), \(i=2,3,4, \dots\) and \(\text{Verdim} (K_{2i})=i\), \(\text{Verdim} (K_{2i+1}) =i+1\), \(i=1,2,3, \dots\).