Let \({\mathcal L}\) be a real linear space of dimension at least 2. We say that a set S of \({\mathcal L}\) is m-triangular convex if it contains at least m points and if m is the smallest integer greater or equal to 3 such that for every m distinct points of S at least one triangle with vertices determined by those points is a subset of S. The author defines also a more general notion of (m,n)-triangular convexity. Here are two of many properties presented in the paper. Every m-triangular convex set is the union of at most m-2 starshaped sets. Every closed connected 4-triangular set is convex.