we-solve (beta)

Math-2305 · mathematics · Analysis

Characterization of cones admitting lifted LMI representations

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Characterize all full-dimensional pointed closed convex cones that are linear images of affine sections of positive semidefinite matrix cones, equivalently cones admitting a lifted linear-matrix-inequality representation. Is this class strictly larger than the class of hyperbolicity cones?

Formulation references: [S1], [S2].

Short English statement

Characterize all full-dimensional pointed closed convex cones that are linear images of affine sections of positive semidefinite matrix cones, equivalently cones admitting a lifted linear-matrix-inequality representation. Is this class strictly larger than the class of hyperbolicity cones?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2305-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2305

A hyperbolicity cone without a semidefinite lift contradicts the proposed universal inclusion. The remaining characterization of all lifted cones is not supplied by an obstruction example.

Characterization of cones admitting lifted LMI representations

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 095; 2026-10-05), the asserted result is that some hyperbolicity cones remain outside the class of projections of finite real semidefinite systems, whatever real coefficients and auxiliary variables are used.

Mathematical statement of the claimed result

Manuscript 1.
The existence claim is a real homogeneous polynomial $p$ and a direction $e$ with $p(e)>0$ such that every univariate $t\mapsto p(te-x)$ is real-rooted, but no finite semidefinite lift represents the resulting closed hyperbolicity cone $K$. Precisely, \[ \nexists N,r<\infty,\ A_i,B_j\in\operatorname{Sym}_N(\mathbb R):\quad K=\{x:\exists y\in\mathbb R^r,\ A_0+\sum_ix_iA_i+\sum_jy_jB_j\succeq0\}. \] This excludes representations even with auxiliary variables, and is stronger than excluding a direct linear matrix inequality.

Outline of the argument

For manuscript 1 (family 095; 2026-10-05), a positive semidefinite Hankel seed with controlled multiplication kernel produces finitely many inverse-limit directions. Separation from Hankel images yields a polynomial patch whose moment deformation contradicts analytic data that any semidefinite lift would require.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 095; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →