Core Variety in Moment Problems

• Core Variety is the terminal set in an iterative geometric construction that determines whether a linear functional admits a representing measure.
• It acts as a complete invariant, ensuring that every finitely atomic representing measure has support contained within this set.
• The construction provides insights into convex-geometric interpretations and determinacy, linking algebraic vanishing with order-theoretic positivity.

Searching arXiv for the primary paper and closely related work on the core variety in moment problems. In the truncated moment problem, the core variety is the terminal set of a descending sequence of zero loci attached to a linear functional $L$ on a finite-dimensional space of functions. In the formulation of Blekherman and Fialkow, one works on a $T_1$ topological space $S$ with a finite-dimensional real vector space $V$ of Borel-measurable functions, and the core variety $\mathcal{CV}(L)$ is obtained by an iterative geometric construction. Its central significance is exact: $L$ has a representing measure if and only if $\mathcal{CV}(L)\neq\varnothing$. When representing measures exist, $L$ has a finitely atomic representing measure, every finitely atomic representing measure has support contained in $\mathcal{CV}(L)$, and the union of the supports of all such measures is precisely $\mathcal{CV}(L)$ (Blekherman et al., 2018). Related work on finite-dimensional spaces of continuous functions identifies the core variety with the set of possible atoms of representing measures and connects it to determinacy (Dio et al., 2017).

1. Ambient setting and basic objects

Let $T_1$0 be a $T_1$1-space, so points are closed, and let $T_1$2 be a finite-dimensional real vector space of Borel-measurable functions on $T_1$3. The dual space $T_1$4 consists of linear functionals $T_1$5. The cone

$T_1$6

collects the functions in $T_1$7 that are nonnegative on $T_1$8. The standing assumptions are that $T_1$9 is full-dimensional in $S$0 and that $S$1 contains at least one strictly positive function $S$2 on $S$3, so $S$4 (Blekherman et al., 2018).

A representing measure for $S$5 is a positive Borel measure $S$6 on $S$7 such that

$S$8

The cone $S$9 denotes all functionals admitting such a measure. By Carathéodory’s theorem, each finitely atomic $V$0 produces an element of $V$1 as a conical combination of point-evaluations $V$2 (Blekherman et al., 2018).

In the classical truncated moment problem, $V$3, the vector space of real $V$4-variable polynomials of degree at most $V$5. The core-variety framework generalizes this by replacing polynomials with an arbitrary finite-dimensional $V$6 of measurable functions on $V$7 (Blekherman et al., 2018).

2. Iterative construction of the core variety

The construction begins with a descending chain of Borel sets

$V$8

defined by

$V$9

$\mathcal{CV}(L)$0

and, for $\mathcal{CV}(L)$1,

$\mathcal{CV}(L)$2

$\mathcal{CV}(L)$3

Here $\mathcal{CV}(L)$4 denotes the common-zero set of a family $\mathcal{CV}(L)$5,

$\mathcal{CV}(L)$6

Because $\mathcal{CV}(L)$7 is finite-dimensional, the chain stabilizes: for some $\mathcal{CV}(L)$8,

$\mathcal{CV}(L)$9

The terminal set

$L$0

is the core variety of $L$1 (Blekherman et al., 2018).

The construction is iterative rather than a single zero-set operation on $L$2. This is a mathematically important point. At each stage, only those kernel elements that are nonnegative on the current set $L$3 are used to define the next set. A plausible implication is that the procedure encodes both algebraic vanishing and order-theoretic positivity, rather than vanishing alone.

An antecedent formulation due to Di Dio and Schmüdgen uses a related descending sequence $L$4 on a locally compact Hausdorff space $L$5, with stabilization in at most $L$6 steps for a finite-dimensional $L$7. The limiting set $L$8 is likewise called the core variety (Dio et al., 2017).

3. Representing measures and the main theorem

For nonzero $L$9, the core-variety theorem gives a dichotomy. Exactly one of the following occurs. Either $\mathcal{CV}(L)\neq\varnothing$0, in which case $\mathcal{CV}(L)\neq\varnothing$1 has no representing measure, or $\mathcal{CV}(L)\neq\varnothing$2. In the latter case, after replacing $\mathcal{CV}(L)\neq\varnothing$3 by $\mathcal{CV}(L)\neq\varnothing$4 if necessary, one may assume $\mathcal{CV}(L)\neq\varnothing$5 for some strictly positive $\mathcal{CV}(L)\neq\varnothing$6. Then $\mathcal{CV}(L)\neq\varnothing$7 admits at least one finitely atomic representing measure; every finitely atomic representing measure $\mathcal{CV}(L)\neq\varnothing$8 representing $\mathcal{CV}(L)\neq\varnothing$9 satisfies $L$0; and the union of the supports of all such measures is exactly $L$1 (Blekherman et al., 2018).

Thus the existence criterion is

$L$2

This gives the core variety the status of a complete invariant for existence of measures in the truncated setting, a formulation stated explicitly in the paper’s remarks (Blekherman et al., 2018).

Under additional topological regularity, the description of supports extends beyond finitely atomic measures. If $L$3 is Hausdorff and the elements of $L$4 are continuous, then $L$5 is also the union of supports of all Radon measures representing $L$6 (Blekherman et al., 2018).

A common misconception is to read the core variety as the support of a particular representing measure. The theorem is stronger and more precise: it is the union of supports of all finitely atomic representing measures, and every such support is contained in it. The remarks also state that $L$7 is the smallest closed subset of $L$8 that contains the supports of all finitely atomic representing measures of $L$9 (Blekherman et al., 2018).

4. Convex-geometric interpretation, atoms, and determinacy

The cone $\mathcal{CV}(L)$0 of functionals with representing measures has a facial decomposition governed by core varieties. For fixed $\mathcal{CV}(L)$1, define

$\mathcal{CV}(L)$2

Then $\mathcal{CV}(L)$3 is a convex face of $\mathcal{CV}(L)$4; one has

$\mathcal{CV}(L)$5

and every face of $\mathcal{CV}(L)$6 arises as $\mathcal{CV}(L)$7 for some $\mathcal{CV}(L)$8. Hence the stratification of $\mathcal{CV}(L)$9 into relative interiors of faces is exactly the stratification by core varieties (Blekherman et al., 2018).

This convex-geometric viewpoint is closely related to earlier work on moment cones. Di Dio and Schmüdgen introduced

$\mathcal{CV}(L)$0

and the set of possible atoms

$\mathcal{CV}(L)$1

They proved that for any nonzero moment functional $\mathcal{CV}(L)$2,

$\mathcal{CV}(L)$3

so the core variety is exactly the set of points that can occur as atoms of some representing measure (Dio et al., 2017).

The same paper relates core variety to determinacy. If $\mathcal{CV}(L)$4 and the chosen functions separate points of $\mathcal{CV}(L)$5, then $\mathcal{CV}(L)$6 is determinate if and only if

$\mathcal{CV}(L)$7

More precisely, $\mathcal{CV}(L)$8 fails to be determinate exactly when one can find more than $\mathcal{CV}(L)$9 distinct atoms in $T_1$00, or equivalently when the vectors $T_1$01 become linearly dependent in $T_1$02 (Dio et al., 2017). This suggests that the size and geometry of the core variety control not only existence of representing measures but also the extent of nonuniqueness.

5. Positive extensions and generalized Riesz–Haviland theorems

The core-variety framework is used to formulate a generalized truncated Riesz–Haviland theorem. Let $T_1$03 be a subspace and $T_1$04. The problem is to determine when $T_1$05 extends to a $T_1$06-positive functional, meaning that there exists $T_1$07 such that $T_1$08 and

$T_1$09

If the set of point-evaluations

$T_1$10

is compact in $T_1$11 — for example, when $T_1$12 is compact and $T_1$13 consists of continuous functions — then the theorem states that if $T_1$14 admits a $T_1$15-positive extension $T_1$16, then in fact

$T_1$17

Equivalently,

$T_1$18

A refinement using compactifications of $T_1$19, via a strictly positive $T_1$20, yields the same conclusion under milder boundedness hypotheses (Blekherman et al., 2018).

The same line of argument extends to a generalized full moment problem. Assume that $T_1$21 is $T_1$22-compact, locally compact Hausdorff, and that $T_1$23 admits an exhaustion

$T_1$24

with each $T_1$25 finite-dimensional. Assume further that for each $T_1$26 there is a strictly positive $T_1$27 such that $T_1$28 for all $T_1$29. Then a linear functional $T_1$30 admits a representing Radon measure on $T_1$31 if and only if

$T_1$32

Moreover, the same statement holds on each truncation $T_1$33, so that a Stochel-type result follows: $T_1$34 admits a representation if and only if each truncation $T_1$35 does (Blekherman et al., 2018).

6. Polynomial case and algorithmic computation

In the classical truncated moment problem one takes

$T_1$36

the real polynomials of degree at most $T_1$37 on $T_1$38. In this setting, each stage of the iteration is algebraic: $T_1$39 Once stabilization occurs at some $T_1$40, one defines the ideal

$T_1$41

Then

$T_1$42

Hence $T_1$43 is a real algebraic variety cut out by the nonnegative-kernel of $T_1$44 (Blekherman et al., 2018).

An iterative algorithmic outline follows the definition directly. One initializes $T_1$45 and iterates. At step $T_1$46, compute a basis $T_1$47 of

$T_1$48

then form

$T_1$49

If $T_1$50, the iteration stops; the final set is $T_1$51 (Blekherman et al., 2018).

The source explicitly notes that computing $T_1$52 may be hard in general. In practice, in the polynomial case one often identifies the sets $T_1$53 via the vanishing of a finite family of SOS-polynomials. A plausible implication is that the computational difficulty is concentrated in identifying the relevant nonnegative kernel elements on each stage, rather than in the finite-dimensional stabilization itself.

7. Scope of the term and terminological distinctions

Within moment theory, “core variety” refers to the iterative zero-set construction attached to a linear functional and its representing measures (Blekherman et al., 2018). The phrase also appears in unrelated literatures with different meanings.

In network analysis, one use concerns a family of core-periphery decompositions generated by scanning a two-parameter family of transition functions $T_1$54, optimizing a core-quality objective by simulated annealing, and aggregating the resulting core vectors into a continuous score $T_1$55 (Rombach et al., 2012). Another use concerns the diversity of hub-and-spoke and layered core-periphery structures, formalized as distinct constrained stochastic block models and compared by Bayesian model selection and minimum description length (Gallagher et al., 2020).

In instruction tuning for LLMs, “core variety” denotes the breadth of distinct activation patterns covered by a selected core set of instruction-response pairs. In that setting, variety is operationalized as coverage over filtered activation tags, and core-set selection is cast as a greedy approximation to a set-cover objective (Bai et al., 29 May 2026).

These usages are different in definition, ambient objects, and mathematical purpose. In the moment-problem literature, the term refers specifically to the set $T_1$56 or $T_1$57, defined from positivity and common zero sets, and used to characterize existence, support, facial structure, and, in related formulations, determinacy of representing measures (Blekherman et al., 2018).