---
title: Core Variety in Moment Problems
url: https://www.emergentmind.com/topics/core-variety
type: topic
---

# Core Variety in Moment Problems

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 \(\mathcal{CV}(L)\) [1804.04276]. 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 [1703.01497].

## 1. Ambient setting and basic objects

Let \(S\) be a \(T_1\)-space, so points are closed, and let \(V\) be a finite-dimensional real vector space of Borel-measurable functions on \(S\). The dual space \(V^*\) consists of linear functionals \(L:V\to\mathbb{R}\). The cone
\[
P=\{\,f\in V:f(s)\ge 0 \text{ for all } s\in S\,\}
\]
collects the functions in \(V\) that are nonnegative on \(S\). The standing assumptions are that \(P\) is full-dimensional in \(V\) and that \(P\) contains at least one strictly positive function \(p>0\) on \(S\), so \(\operatorname{int} P\neq\varnothing\) [1804.04276].

A **representing measure** for \(L\in V^*\) is a positive Borel measure \(\mu\) on \(S\) such that
\[
L(f)=\int_S f\,d\mu,\qquad \forall f\in V.
\]
The cone \(M\subset V^*\) denotes all functionals admitting such a measure. By Carathéodory’s theorem, each finitely atomic \(\mu\) produces an element of \(M\) as a conical combination of point-evaluations \(L_s(f)=f(s)\) [1804.04276].

In the classical truncated moment problem, \(V=\mathcal{P}_d(\mathbb{R}^n)\), the vector space of real \(n\)-variable polynomials of degree at most \(d\). The core-variety framework generalizes this by replacing polynomials with an arbitrary finite-dimensional \(V\) of measurable functions on \(S\) [1804.04276].

## 2. Iterative construction of the core variety

The construction begins with a descending chain of Borel sets
\[
S_0\supseteq S_1\supseteq S_2\supseteq \cdots
\]
defined by
\[
S_0:=S,
\]
\[
S_1:=Z(\,p\in P:L(p)=0\,)
=\{\,s\in S:p(s)=0 \text{ for every } p\in P\cap \ker L\,\},
\]
and, for \(k\ge 1\),
\[
S_{k+1}:=Z(\,f\in V:L(f)=0 \text{ and } f(s)\ge 0 \ \forall s\in S_k\,)
\]
\[
=\{\,s\in S_k:f(s)=0 \text{ for every } f\in V\cap \ker L \text{ with } f\ge 0 \text{ on } S_k\,\}.
\]
Here \(Z(T)\) denotes the common-zero set of a family \(T\subset V\),
\[
Z(T):=\{\,s\in S:t(s)=0 \text{ for all } t\in T\,\}.
\]
Because \(V\) is finite-dimensional, the chain stabilizes: for some \(k\le \dim V-1\),
\[
S_k=S_{k+1}=S_{k+2}=\cdots.
\]
The terminal set
\[
\mathcal{CV}(L):=S_k
\]
is the **core variety** of \(L\) [1804.04276].

The construction is iterative rather than a single zero-set operation on \(\ker L\). This is a mathematically important point. At each stage, only those kernel elements that are nonnegative on the current set \(S_k\) 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 \(V_0(L)\supseteq V_1(L)\supseteq \cdots\) on a locally compact Hausdorff space \(X\), with stabilization in at most \(\dim\mathcal{F}\) steps for a finite-dimensional \(\mathcal{F}\subset C(X;\mathbb{R})\). The limiting set \(\mathcal{V}_c(L)\) is likewise called the core variety [1703.01497].

## 3. Representing measures and the main theorem

For nonzero \(L\in V^*\), the core-variety theorem gives a dichotomy. Exactly one of the following occurs. Either \(\mathcal{CV}(L)=\varnothing\), in which case \(L\) has no representing measure, or \(\mathcal{CV}(L)\neq\varnothing\). In the latter case, after replacing \(L\) by \(-L\) if necessary, one may assume \(L(p)>0\) for some strictly positive \(p\in P\). Then \(L\) admits at least one finitely atomic representing measure; every finitely atomic representing measure \(\mu\) representing \(L\) satisfies \(\operatorname{supp}\mu\subseteq \mathcal{CV}(L)\); and the union of the supports of all such measures is exactly \(\mathcal{CV}(L)\) [1804.04276].

Thus the existence criterion is
\[
L \text{ has a representing measure } \Longleftrightarrow \mathcal{CV}(L)\neq\varnothing.
\]
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 [1804.04276].

Under additional topological regularity, the description of supports extends beyond finitely atomic measures. If \(S\) is Hausdorff and the elements of \(V\) are continuous, then \(\mathcal{CV}(L)\) is also the union of supports of all Radon measures representing \(L\) [1804.04276].

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 \(\mathcal{CV}(L)\) is the smallest closed subset of \(S\) that contains the supports of all finitely atomic representing measures of \(L\) [1804.04276].

## 4. Convex-geometric interpretation, atoms, and determinacy

The cone \(M\subset V^*\) of functionals with representing measures has a facial decomposition governed by core varieties. For fixed \(L\in M\), define
\[
F_L:=\{\,m\in M:\mathcal{CV}(m)\subseteq \mathcal{CV}(L)\,\}.
\]
Then \(F_L\) is a convex face of \(M\); one has
\[
m\in \operatorname{relint} F_L \Longleftrightarrow \mathcal{CV}(m)=\mathcal{CV}(L);
\]
and every face of \(M\) arises as \(F_L\) for some \(L\in M\). Hence the stratification of \(M\) into relative interiors of faces is exactly the stratification by core varieties [1804.04276].

This convex-geometric viewpoint is closely related to earlier work on moment cones. Di Dio and Schmüdgen introduced
\[
N_+(L):=\{\,f\in \mathcal{F}^+:L(f)=0\,\},
\qquad
V_+(L):=\{\,x\in X:f(x)=0 \ \forall f\in N_+(L)\,\},
\]
and the set of possible atoms
\[
W(L):=\{\,x\in X:\exists\,\mu\in M_L \text{ with } \mu(\{x\})>0\,\}.
\]
They proved that for any nonzero moment functional \(L\),
\[
W(L)=\mathcal{V}_c(L),
\]
so the core variety is exactly the set of points that can occur as atoms of some representing measure [1703.01497].

The same paper relates core variety to **determinacy**. If \(X\subset\mathbb{R}^n\) and the chosen functions separate points of \(X\), then \(L\) is determinate if and only if
\[
|\mathcal{V}_c(L)|\le \dim\mathcal{F}.
\]
More precisely, \(L\) fails to be determinate exactly when one can find more than \(\dim\mathcal{F}\) distinct atoms in \(\mathcal{V}_c(L)\), or equivalently when the vectors \(\{s_F(x):x\in\mathcal{V}_c(L)\}\) become linearly dependent in \(\mathbb{R}^m\) [1703.01497]. 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 \(U\subseteq V\) be a subspace and \(L\in U^*\). The problem is to determine when \(L\) extends to a \(V\)-positive functional, meaning that there exists \(\bar L\in V^*\) such that \(\bar L|_U=L\) and
\[
\bar L(f)\ge 0 \qquad \text{for all } f\in P.
\]
If the set of point-evaluations
\[
S^*=\{L_s:s\in S\}
\]
is compact in \(V^*\) — for example, when \(S\) is compact and \(V\) consists of continuous functions — then the theorem states that if \(L\) admits a \(V\)-positive extension \(\bar L\), then in fact
\[
L=\sum_{i=1}^N a_i L_{s_i}, \qquad a_i>0,\ s_i\in S.
\]
Equivalently,
\[
L \text{ has a positive extension } \Longleftrightarrow L \text{ is a nonnegative combination of point-evaluations.}
\]
A refinement using compactifications of \(S^*\), via a strictly positive \(p\in \operatorname{int}P\), yields the same conclusion under milder boundedness hypotheses [1804.04276].

The same line of argument extends to a generalized full moment problem. Assume that \(S\) is \(\sigma\)-compact, locally compact Hausdorff, and that \(V\subset C_0(S)\) admits an exhaustion
\[
V_1\subseteq V_2\subseteq \cdots,\qquad \bigcup_j V_j=V,
\]
with each \(V_j\) finite-dimensional. Assume further that for each \(j\ge 2\) there is a strictly positive \(p_j\in V_j\) such that \(q/p_j\in C_0(S)\) for all \(q\in V_{j-1}\). Then a linear functional \(L:V\to\mathbb{R}\) admits a representing Radon measure on \(S\) if and only if
\[
L(f)\ge 0 \qquad \text{for all } f\in P.
\]
Moreover, the same statement holds on each truncation \(V_j\), so that a Stochel-type result follows: \(L\) admits a representation if and only if each truncation \(L|_{V_j}\) does [1804.04276].

## 6. Polynomial case and algorithmic computation

In the classical truncated moment problem one takes
\[
V=\mathcal{P}_d(\mathbb{R}^n),
\]
the real polynomials of degree at most \(d\) on \(S\subseteq \mathbb{R}^n\). In this setting, each stage of the iteration is algebraic:
\[
S_{k+1}=\{\,x\in S_k : p(x)=0 \ \forall\, p\in \ker L \text{ with } p\ge 0 \text{ on } S_k\,\}.
\]
Once stabilization occurs at some \(k\), one defines the ideal
\[
I:=\langle\, p(x): p\in \ker L,\ p\ge 0 \text{ on } S_k\,\rangle \subset \mathbb{R}[x_1,\dots,x_n].
\]
Then
\[
\mathcal{CV}(L)=V_{\mathbb{R}}(I):=\{\,x\in \mathbb{R}^n: f(x)=0 \ \forall f\in I\,\}.
\]
Hence \(\mathcal{CV}(L)\) is a real algebraic variety cut out by the nonnegative-kernel of \(L\) [1804.04276].

An iterative algorithmic outline follows the definition directly. One initializes \(S_0:=S\) and iterates. At step \(k\), compute a basis \(\{f_1,\dots,f_m\}\) of
\[
T_k=\{\,f\in V:L(f)=0 \text{ and } f(s)\ge 0 \ \forall s\in S_k\,\},
\]
then form
\[
S_{k+1}=\{\,s\in S_k:f_i(s)=0 \text{ for all } i=1,\dots,m\,\}.
\]
If \(S_{k+1}=S_k\), the iteration stops; the final set is \(\mathcal{CV}(L)\) [1804.04276].

The source explicitly notes that computing \(\mathcal{CV}(L)\) may be hard in general. In practice, in the polynomial case one often identifies the sets \(S_k\) 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 [1804.04276]. 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 \((\alpha,\beta)\), optimizing a core-quality objective by simulated annealing, and aggregating the resulting core vectors into a continuous score \(CS(i)\) [1202.2684]. 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 [2005.10191].

In instruction tuning for large language models, “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 [2605.30857].

These usages are different in definition, ambient objects, and mathematical purpose. In the moment-problem literature, the term refers specifically to the set \(\mathcal{CV}(L)\) or \(\mathcal{V}_c(L)\), defined from positivity and common zero sets, and used to characterize existence, support, facial structure, and, in related formulations, determinacy of representing measures [1804.04276].

Source: https://www.emergentmind.com/topics/core-variety