Conditional graph entropy is known to be the minimal rate for a natural functional compression problem with side information at the receiver. In this paper we show that it can be formulated as an alternating minimization problem, which gives rise to a simple iterative algorithm for numerically computing (conditional) graph entropy. This also leads to a new formula which shows that conditional graph entropy is part of a more general framework: the solution of an optimization problem over a convex corner. In the special case of graph entropy (i.e., unconditioned version) this was known due to Csiszár, Körner, Lovász, Marton, and Simonyi. In that case the role of the convex corner was played by the so-called vertex packing polytope. In the conditional version it is a more intricate convex body but the function to minimize is the same. Furthermore, we describe a dual problem that leads to an optimality check and an error bound for the iterative algorithm.

A finite set [Formula: see text] in [Formula: see text] is called an acute set if any angle determined by three points of [Formula: see text] is acute. We examine the maximal cardinality [Formula: see text] of a [Formula: see text]-dimensional acute set. The exact value of [Formula: see text] is known only for [Formula: see text]. For each [Formula: see text] we improve on the best known lower bound for [Formula: see text]. We present different approaches. On one hand, we give a probabilistic proof that [Formula: see text]. (This improves a random construction given by Erdo˝s and Füredi.) On the other hand, we give an almost exponential constructive example which outdoes the random construction in low dimension ([Formula: see text]). Both approaches use the small dimensional examples that we found partly by hand ([Formula: see text], 5) and partly by computer ([Formula: see text]). We also investigate the following variant of the above problem: what is the maximal size [Formula: see text] of a [Formula: see text]-dimensional cubic acute set (that is, an acute set contained in the vertex set of a [Formula: see text]-dimensional hypercube)? We give an almost exponential constructive lower bound, and we improve on the best known upper bound.