Abstract. We investigate the following question: How close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal distance between such polytopes appears in complexity bounds of optimization algorithms. We provide nearly matching bounds on this distance and discuss its exact computation. We also give similar bounds for disjoint rational polytopes whose binary encoding length is prescribed.

We introduce and study the problem of optimizing arbitrary functions over degree sequences of hypergraphs and multihypergraphs. We show that over multihypergraphs the problem can be solved in polynomial time. For hypergraphs, we show that deciding whether a given sequence is the degree sequence of a 3-hypergraph is NP-complete, thereby solving a 30 year long open problem. This implies that optimization over hypergraphs is hard even for simple concave functions. In contrast, we show that for graphs, if the functions at vertices are the same, then the problem is polynomial time solvable. We also provide positive results for convex optimization over multihypergraphs and graphs and exploit connections to degree sequence polytopes and threshold graphs. We then elaborate on connections to the emerging theory of shifted combinatorial optimization.

The colourful simplicial depth conjecture states that any point in the convex hull of each of $d+1$ sets, or colours, of $d+1$ points in general position in $\mathbb{R}^d$ is contained in at least $d^2+1$ simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which cannot arise from a colourful point configuration. The number of octahedral systems is also given.