We construct the first known hollow lattice polytopes of width larger than dimension: a hollow lattice polytope (resp. a hollow lattice simplex) of dimension 14 (resp. 404) and of width 15 (resp. 408). We also construct a hollow (non-lattice) tetrahedron of width 2 + √ 2 and conjecture that this is the maximum width among 3-dimensional hollow convex bodies.We show that the maximum lattice width grows (at least) additively with d. In particular, the constructions above imply the existence of hollow lattice polytopes (resp. hollow simplices) of arbitrarily large dimension d and width 1.14d (resp. 1.01d).

A relative simplicial complex is a collection of sets of the form ∆ \ Γ, where Γ ⊂ ∆ are simplicial complexes. Relative complexes have played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of f -vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates.Moreover, we characterize h-vectors of fully Cohen-Macaulay relative complexes as well as hvectors of Cohen-Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Björner.

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ n ≥ d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.

We show that the following classes of lattice polytopes have unimodular covers, in dimension three: the class of parallelepipeds, the class of centrally symmetric polytopes, and the class of Cayley sums Cay(P, Q) where the normal fan of Q refines that of P . This improves results of Beck et al. (2018) and Haase et al. (2008) where the last two classes were shown to be IDP.

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