bibkey: carlisilva2026kfoldtheta
authors: Marcel K Carli Silva; Gabriel Coutinho; Thiago Oliveira; Levent Tunçel
year: 2026
title: On some k-fold generalizations of Lovász theta and their sandwich theorems
doi: null
url: https://arxiv.org/html/2609.06312v1
claim: Theorem 16 defines the trace-k orthonormal representation set; Section 7 item (iv) asks whether it is convex and separately asks for its closed convex lower hull.
strata_touched:
  - D5/S3/Combinatorics/Orthogonality/ThetaRepresentationNonconvex
license: citation-only
triage: anchor

The trace-k orthonormal representation set

Verified locator

https://arxiv.org/html/2609.06312v1

Definition: Section 5.3, proof of Theorem 16. Question: Section 7, Open questions, item (iv).

Definition and scope

For a finite simple graph on V, the set consists of nonnegative vectors x for which a common real symmetric positive semidefinite matrix M satisfies I−M positive semidefinite and tr(M)=k, and a common family u:V→R^V satisfies u_i·u_i=1, u_i·u_j=0 on edges of G, and x_i=u_i·Mu_i. The paper calls u an orthonormal representation of the complement of G. Vectors at nonadjacent vertices may coincide. The definition allows all positive semidefinite contractions of exact trace k, not only projectors.

The exact question is: “Is the orthonormal representation set M_k(G) convex? If not, can one characterize its closed convex lower hull, i.e., the convex corner whose support function is φ_k(G,·)?”

The convexity assertion and the lower hull characterization are distinct. The five-cycle at k=2 is in the integer regime and in the positive feasible real regime 0<k≤|V|. No claim about the lower hull is part of the associated nonconvexity theorem.