Compact-set structure in free locally r-convex spaces

Characterize the compact subsets of L^{(r)}(X) for 0\le r<1, and determine in particular whether, for 0<r<1, every compact subset of L^{(r)}(X) is contained in a set of the form $\{\sum_j c_j y_j:\sum_j |c_j|^r\le n\}$ for some n.

Background

The paper establishes that compact sets in L^{(r)}(X) can behave differently from compact sets in V(X). In particular, for 0\le r<1, it gives examples of compact sets in L^{(r)}(X) whose supports have unbounded cardinality, whereas compact subsets of V(X) have uniformly bounded support length.

The authors ask for a general description of compact subsets in L^{(r)}(X). The proposed condition would bound the r-power sum of coefficients uniformly over each compact set, extending the role played by small quasinorm balls in the hemicompactness arguments.