Summary: Let \(S\) be a set of \(n\) points in the plane, and let \(DT (S)\) be the planar graph of the Delaunay triangulation of \(S\). For a pair of points \(a, b \in S\), denote by \(|{ab}|\) the Euclidean distance between \(a\) and \(b\). Denote by \(DT (a, b)\) the shortest path in \(DT (S)\) between \(a\) and \(b\), and let \(|DT (a, b)|\) be the total length of \(DT (a, b)\). D. P. Dobkin et al. [Discrete Comput. Geom. 5, No. 4, 399–407 (1990; Zbl 0693.05045)] were the first to show that \(DT (S)\) can be used to approximate the complete graph of \(S\) in the sense that the stretch factor \(\frac{|DT(a, b)|}{|a b|}\) is upper bounded by \(((1 + \sqrt{5})/2) \pi \approx 5.08\). Recently, Xia improved this factor to 1.998. A. Biniaz et al. [J. Comput. Geom. 7, No. 1, 520–539 (2016; Zbl 1405.68399)] have also shown that if the points of \(S\) are in convex position (i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of \(S\) are in convex position, then the stretch factor of \(DT (S)\) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case.