Convex Overlap-Based Fidelity

• The paper establishes that convex overlap-based fidelity quantifies similarity by maximizing overlap (area, trace, etc.) and leverages convexity for tractable, concave optimization.
• It demonstrates that in geometric settings the concavity of the square-root overlap function enables exact linear-time algorithms while in quantum settings fidelity aligns with convex state spaces.
• The framework extends across disciplines, including causal inference and lattice theory, showing that convex formulations yield robust, reliable measures of similarity.

Searching arXiv for the cited papers and closely related overlap/fidelity work. arxiv_search(query="(Chan et al., 25 Apr 2025) OR (Li et al., 2015) OR (Paiva et al., 2019) OR (Zhou et al., 2021) OR (Ghosh, 2018) OR (Kweon et al., 2023) OR (Kweon et al., 2023)", max_results=10, sort_by="relevance") Convex Overlap-Based Fidelity denotes a class of fidelity or similarity constructions in which closeness is measured by overlap, while convexity enters through the objects being compared, the admissible mixtures or transformations, or the optimization landscape itself. In computational geometry, the canonical example is the overlap area

$A(t):=\textnormal{Area}((P+t)\cap Q)$

of two convex polygons under translation, with fidelity maximized by

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$

and with the key structural property that $\sqrt{A(t)}$ is downward concave on the region where $A(t)>0$ (Chan et al., 25 Apr 2025). In quantum information, fidelity is an overlap quantity on density matrices,

$$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$

and its geometry is analyzed on the closed, convex set of states and on convex sets of channels (Li et al., 2015). Related formulations appear in convex polygon matching (Kweon et al., 2023), convex-hull overlap in causal inference (Ghosh, 2018), lattice-valued overlap aggregation (Paiva et al., 2019), optimal convex approximation of quantum states (Zhou et al., 2021), overlap functionals turned into convex programs (White, 2022), and smooth convex data fidelities built from linearly involved overlap operators in inverse problems (Yata et al., 3 Sep 2025). Taken together, these works define a coherent research perspective rather than a single universal formula.

1. Core formulations and recurring mathematical structure

The cited literature uses “overlap” in several distinct senses, but the underlying pattern is stable: a fidelity is large when two objects align well, when a target lies near a convex set of feasible representations, or when a design remains in a shared support region. In convex shape matching, overlap is literal intersection area or volume under translation (Chan et al., 25 Apr 2025). In quantum theory, fidelity is an overlap-like trace functional on density matrices and reduces to the modulus of an inner product for pure states (Li et al., 2015). In lattice theory, overlap is a bivariate aggregation operator on a bounded lattice, with symmetry, boundary conditions, and monotonicity built into the definition (Paiva et al., 2019).

Setting	Overlap object	Convex structure
Convex polygons	$\textnormal{Area}((P+t)\cap Q)$	Convex shapes; concavity of $\sqrt{A(t)}$ (Chan et al., 25 Apr 2025)
Quantum states	$F(\rho,\sigma)$	Convex set of density matrices and channels (Li et al., 2015)
Convex approximation	$\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$	Convex hull of available states (Zhou et al., 2021)
Causal inference	$$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$	Convex hull overlap of treatment groups (Ghosh, 2018)
Lattice aggregation	$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$0	Bounded lattices; convex sum via t-norm/t-conorm (Paiva et al., 2019)

A common structural feature is that overlap is often easier to optimize or characterize because convexity suppresses pathological combinatorics. In the geometric setting, convexity yields a well-behaved configuration-space objective. In the quantum setting, concavity of fidelity in each argument ties extremal behavior to pure states or extreme channels. In causal inference, convex hull intersection converts positivity into a geometric separability question. This suggests that convex overlap-based fidelity is less a domain-specific definition than a transferable analytic template.

2. Convex polygons under translation as the canonical geometric model

The most explicit formulation appears in Chan and Hair’s treatment of the maximum overlap problem for convex polygons. Given convex polygons $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$1 with $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$2 and $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$3 edges, the objective is

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$4

with $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$5 and

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$6

The central imported lemma from de Berg et al. is

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$7

which places the problem in the regime of fixed-dimensional concave maximization (Chan et al., 25 Apr 2025).

The same work explicitly interprets overlap as a fidelity criterion. It states that the maximum overlap problem may be viewed as shape matching, and that overlap area is “another very natural measure” because “the larger it is, the closer the two shapes are” (Chan et al., 25 Apr 2025). Under this interpretation,

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$8

is a positive similarity measure: it vanishes when the translated shapes do not intersect and increases with alignment quality.

The technical structure of $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$9 is unusually favorable. For each edge pair $\sqrt{A(t)}$0, $\sqrt{A(t)}$1, the set of translations for which $\sqrt{A(t)}$2 intersects $\sqrt{A(t)}$3 is a parallelogram $\sqrt{A(t)}$4 in translation space. The union of their boundaries,

$\sqrt{A(t)}$5

partitions configuration space into cells. On any triangle or segment $\sqrt{A(t)}$6 contained in one cell of $\sqrt{A(t)}$7, the overlap area becomes a constant-complexity quadratic: $\sqrt{A(t)}$8 This piecewise-quadratic structure is the local algebraic form of geometric overlap fidelity (Chan et al., 25 Apr 2025).

Chan and Hair’s principal algorithmic result is the first randomized exact algorithm with expected linear running time: $\sqrt{A(t)}$9 The method combines configuration-space geometry, local quadratic formulas, block partitioning by edge angle, $A(t)>0$0-block structures, randomized cuttings, cascade refinements, and multidimensional parametric search. Two key oracle costs are

$A(t)>0$1

and

$A(t)>0$2

The final bound is

$A(t)>0$3

improving the earlier $A(t)>0$4 exact algorithm of de Berg, Cheong, Devillers, van Kreveld, and Teillaud (Chan et al., 25 Apr 2025).

This geometric model establishes the most concrete meaning of convex overlap-based fidelity: overlap is the fidelity; convexity of the shapes and concavity of $A(t)>0$5 make the fidelity landscape tractable; and exact optimization is possible in linear expected time.

3. Multi-shape overlap, higher-dimensional configurations, and sets of maximizers

The same overlap perspective extends beyond two polygons. For constant $A(t)>0$6, the overlap area of $A(t)>0$7 convex polygons under independent translations is

$A(t)>0$8

defined on the quotient configuration space

$A(t)>0$9

which factors out global translation (Kweon et al., 2023). The support of $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$0 is compact, $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$1 is continuous, and $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$2 is concave on its support. On regions where no event polytope intersects the interior, the intersection polygon has fixed combinatorics and the shoelace formula shows that $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$3 is a quadratic polynomial in the translation parameters (Kweon et al., 2023).

For fixed $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$4, the maximization problem admits an

$$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$5

algorithm, and once one maximizer is known, the full set

$$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$6

can be computed in $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$7 time as a convex polytope given by $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$8 linear constraints (Kweon et al., 2023). A crucial structural fact is that all maximal overlaps are translates of a canonical maximal overlap polygon $$F(\rho,\sigma)=\operatorname{Tr}\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr),$$9, and that

$\textnormal{Area}((P+t)\cap Q)$0

This gives an explicit convex description of the entire maximizer set.

A related three-dimensional variant considers a convex polyhedron $\textnormal{Area}((P+t)\cap Q)$1 and a convex polygon $\textnormal{Area}((P+t)\cap Q)$2 translated in $\textnormal{Area}((P+t)\cap Q)$3, with objective

$\textnormal{Area}((P+t)\cap Q)$4

The overlap function is continuous and piecewise quadratic on the interior of its support, and $\textnormal{Area}((P+t)\cap Q)$5 is concave on its support by a Brunn–Minkowski argument (Kweon et al., 2023). The configuration space is partitioned by event polygons of three kinds: $\textnormal{Area}((P+t)\cap Q)$6 for a vertex $\textnormal{Area}((P+t)\cap Q)$7 of $\textnormal{Area}((P+t)\cap Q)$8, $\textnormal{Area}((P+t)\cap Q)$9 for a face $\sqrt{A(t)}$0 of $\sqrt{A(t)}$1 and a vertex $\sqrt{A(t)}$2 of $\sqrt{A(t)}$3, and $\sqrt{A(t)}$4 for edges $\sqrt{A(t)}$5 of $\sqrt{A(t)}$6 and $\sqrt{A(t)}$7 of $\sqrt{A(t)}$8. A deterministic algorithm finds a maximizing translation in

$\sqrt{A(t)}$9

time (Kweon et al., 2023).

The same paper applies the method to the maximum overlap of three convex polygons and to minimization of symmetric difference under homothety. For convex polygons $F(\rho,\sigma)$0, the symmetric difference satisfies

$F(\rho,\sigma)$1

and for fixed areas minimizing $F(\rho,\sigma)$2 is equivalent to maximizing $F(\rho,\sigma)$3. This places overlap-based fidelity and symmetric-difference dissimilarity in direct duality (Kweon et al., 2023). A plausible implication is that overlap-based fidelity is especially natural when area or volume is preserved, while symmetric-difference formulations are better adapted to scale changes.

4. Quantum fidelity, convex sets of states, and convex approximation

In quantum information, fidelity is introduced algebraically but analyzed geometrically. For density matrices $F(\rho,\sigma)$4 and $F(\rho,\sigma)$5,

$F(\rho,\sigma)$6

and for pure states it reduces to overlap: $F(\rho,\sigma)$7 The paper on geometric interpretations of quantum fidelity emphasizes that the state space $F(\rho,\sigma)$8 is a compact, closed convex set, with pure states as extreme points and $F(\rho,\sigma)$9 as the maximally mixed center (Li et al., 2015).

Fidelity is concave in each argument: $\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$0 and this concavity drives several convex-geometric interpretations. If $\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$1 contains all pure states, then

$\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$2

For quantum channels, minimizing over all CPTP maps gives

$\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$3

while minimization over unital or mixed unitary channels yields

$\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$4

the “anti-aligned eigenvalues” expression (Li et al., 2015). Here $\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$5 is interpreted as a convex combination coefficient measuring how close $\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$6 is to the maximally mixed state versus the boundary of singular density matrices.

The same paper gives an explicitly geometric overlap interpretation for subspaces. If

$\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$7

are normalized projections onto subspaces $\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$8, then in the commuting case

$\max_{\{p_i\}}F(\rho,\sum_i p_i\rho_i)$9

and in general

$$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$0

where $$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$1 are the canonical angles and $$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$2 (Li et al., 2015). Thus fidelity becomes a scaled overlap of principal directions.

A second quantum line of work studies optimal convex approximations. Given a target state $$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$3 and available states $$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$4, the fidelity-based approximation problem is

$$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$5

For qubits and specific available sets such as the six Pauli eigenstates $$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$6, complete exact solutions are obtained via Bloch-vector geometry and KKT conditions (Zhou et al., 2021). In the region

$$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$7

the target state is completely represented by the convex hull of available states and

$$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$8

Outside this region, optimal mixtures live on faces or edges of the octahedral convex hull, so the maximizing mixture typically uses only two or three states (Zhou et al., 2021). This is convex overlap-based fidelity in the strict sense of maximizing overlap with a convex set of feasible states.

5. Overlap as convex program, design criterion, and aggregation law

The overlap viewpoint also appears where the primary object is not a geometric intersection area. Erdős’ minimum overlap problem studies partitions $$\operatorname{co}(\mathbf{Z}_0)\cap \operatorname{co}(\mathbf{Z}_1)$$9 with $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$00, difference counts

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$01

and the asymptotic constant

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$02

In its continuous form, one considers measurable $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$03 with $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$04, its complement $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$05, and the overlap functional

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$06

so that

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$07

The paper translates this into a convex optimization program using moment identities, Fourier coefficients, interval averages, and second-order cone constraints, proving

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$08

(White, 2022). Here overlap becomes a cross-correlation-type functional and convexity arises at the level of the feasible program.

In causal inference, Ghosh defines relaxed covariate overlap through convex hulls of treated and control covariates,

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$09

with overlap condition

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$10

The distance between hulls is obtained by minimizing

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$11

over simplex weights, and failure of overlap is equivalent to the existence of a separating hyperplane, linking convex hull geometry to the margin of a linear SVM (Ghosh, 2018). This is not a fidelity in the trace or area sense, but it is an overlap-based notion of design fidelity: causal comparisons are regarded as credible only in the shared convex support region.

In lattice theory, overlap functions generalize from $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$12 to bounded lattices $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$13. An $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$14-overlap

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$15

satisfies symmetry, the zero boundary

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$16

the one boundary

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$17

and monotonicity; continuity distinguishes overlaps from quasi-overlaps (Paiva et al., 2019). A lattice-valued convex combination is built through a t-norm $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$18, a t-conorm $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$19, and weights $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$20 with $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$21: $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$22 Under the stated hypotheses, convex sums of overlaps or quasi-overlaps are again overlaps or quasi-overlaps (Paiva et al., 2019). This provides an abstract aggregation-theoretic meaning of convex overlap-based fidelity.

6. Scope, limitations, and adjacent developments

Several misconceptions are explicitly ruled out by the cited works. First, overlap-based fidelity is not synonymous with distance. In the geometric papers, overlap is a positive similarity, whereas Hausdorff distance and Fréchet distance are negative dissimilarities (Chan et al., 25 Apr 2025). In quantum theory, fidelity and trace distance induce different optimal convex approximations; for many qubit regimes, the fidelity-based optimum is closer to the target state than the trace-norm-based optimum, but the two criteria do not coincide (Zhou et al., 2021). In causal inference, convex hull overlap is not the same as scalar propensity-score overlap, because it is multivariate and geometric rather than one-dimensional (Ghosh, 2018).

Second, convexity is essential to most exact results. The linear-time algorithm of Chan and Hair is specific to convex polygons under translation and does not directly handle rotation or non-convexity (Chan et al., 25 Apr 2025). For several convex polygons, $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$23 is assumed constant (Kweon et al., 2023). For the polyhedron–polygon problem and the homothety problem, the arguments rely on event structures, cuttings, and Brunn–Minkowski-type concavity that are not available in the same form for general non-convex shapes (Kweon et al., 2023). Lattice-overlap theory changes substantially when continuity is dropped, which motivates quasi-overlaps (Paiva et al., 2019).

Third, the overlap perspective extends naturally into inverse problems with smooth convex data fidelities. The LiGME-SCF framework considers objectives of the form

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$24

where $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$25 is a smooth convex data fidelity, $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$26 is a linear observation operator, and $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$27 is a generalized Moreau enhanced nonconvex regularizer. Overall convexity is guaranteed by

$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$28

equivalently by the requirement that $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$29 be $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$30-strongly convex relative to $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$31 (Yata et al., 3 Sep 2025). The paper treats overlapped measurements as encoded by the linear operator $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$32, so that a convex overlap-based fidelity may take the form $$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$33 with nonquadratic choices such as Poisson and clipping likelihoods. This suggests a broader interpretation in which “overlap” is represented by linearly shared measurements rather than set intersection.

The main open frontiers named across the literature are consistent. Exact linear-time behavior is currently established in the basic two-dimensional convex-translation case (Chan et al., 25 Apr 2025). Higher dimensions, rigid motions, non-convex shapes, and large-$$\max_{t\in\mathbb{R}^2}\;\textnormal{Area}((P+t)\cap Q),$$34 multi-object matching remain harder (Kweon et al., 2023). In quantum settings, explicit exact formulas are presently strongest for qubits and particular available sets (Zhou et al., 2021). In causal inference, the geometric margin identifies a high-fidelity subpopulation but makes the estimand data-adaptive (Ghosh, 2018). These limitations do not weaken the concept; they delimit the settings in which convex overlap-based fidelity is presently mathematically sharp and algorithmically effective.