Geometry of good sets inn-fold Cartesian product

Abstract

We propose here a multidimensional generalisation of the notion of link introduced in our previous papers and we discuss some consequences for simplicial measures and sums of function algebras.

Some further remarks on good sets

“…In p. 190 of [4], the question whether full components are the same as related components for n ≥ 3, is raised. Theorem 1 of [2] answers the question partially, where it is proved that a full set with finitely many related components is itself related.…”

“…Let S be a related set in n-fold cartesian product X 1 × X 2 × · · · × X n . Given two points x, y ∈ S the smallest related subset of S containing x and y is called the geodesic between x and y ( [4], p. 190). In [5], it is shown that if there is an upper bound for lengths of geodesics in S, then the solutions u 1 , .…”

“…A subset S ⊂ is said to be full, if S is a maximal good set in 1 S × 2 S × · · · × n S. A set S ⊂ is full if and only if it has a boundary consisting of n − 1 points ( [4], Theorem 3, page 185).…”

“…If a set S is good, maximal full subsets of S form a partition of S. They are called full components of S ( [4], p. 183).…”

On the support of extremal martingale measures with given marginals: the countable case

“…Many papers followed, e.g. [5], [6], [12], [14], [17], [22], [23], [25], [27], and [28] among others, giving different kinds of characterizations in the finite or countable case and going from functional analysis to combinatorics. In particular, Denny [12] extended to the countable case Birkhoff's result about absence of cycles in the support of extremal measures.…”