We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $\mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$-vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that contains every $n$-vertex forest as a subgraph. The upper bound of $O\!\left(n \log n\right)$ edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that every $n$-vertex convex geometric graph that is universal for $n$-vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega _h(n^{2-1/h})$, for every positive integer $h$; this almost matches the trivial $O(n^2)$ upper bound given by the $n$-vertex complete convex geometric graph. Finally, we prove that there exists an $n$-vertex convex geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$-vertex caterpillars.

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n ). This improves an earlier bound of O(1.6181n ) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n ) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3-space, and in general in d dimensions.

1. (i) The number of tetrahedra of minimum (non-zero) volume spanned by n points in 3 is at most , and there are point sets for which this number is . We also present an O(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every , the maximum number of k-dimensional simplices of minimum (non-zero) volume spanned by n points in d is Θ(nk).

2. (ii) The number of unit volume tetrahedra determined by n points in 3 is O(n7/2), and there are point sets for which this number is Ω(n3 log logn).

3. (iii) For every , the minimum number of distinct volumes of all full-dimensional simplices determined by n points in d, not all on a hyperplane, is Θ(n).