Lowest Centroid: Extremal Optimization

• Lowest centroid is an extremal notion defined by optimizing centroids under geometric, combinatorial, or physical constraints across diverse fields.
• In convex geometry, centroid localization (e.g., the sharp 4/21 bound) informs affine comparison metrics like the centroid-constrained Banach–Mazur distance.
• In clustering and tree theory, lowest centroid methods yield optimal data initialization and minimal depth nodes, thereby enhancing performance and structural analysis.

Searching arXiv for recent and relevant papers on “lowest centroid” and closely related centroid notions.

“Lowest centroid” is not a single standardized term. Across convex geometry, clustering, random tree theory, and mechanics, it denotes an extremal centroid notion: a centroid constrained to a smallest admissible region, a centroid seed obtained from a lowest-energy optimization, a pseudo-centroid minimizing a clusterwise span, the centroid nearest the root in a tree, a centroid body with minimal admissible volume, or the lowest vertical position of a center of gravity during filling (Lassak, 2022, Lassak, 2022, Allgood et al., 2024, Glover, 2016, Durant et al., 2018, Haddad et al., 2019, Sjödin, 2017). The common structure is variational: a centroid is identified not merely as a mean, but as an optimizer under geometric, combinatorial, or physical constraints.

1. Terminological range and basic meanings

The term “centroid” changes meaning with the underlying object. In Euclidean convexity it is the barycenter of a body; in prototype-based clustering it is a representative or seed; in coordinate-free clustering it may be a data point minimizing a span functional; in tree theory it is a node minimizing total graph distance; in $L_p$ convex geometry it gives rise to a derived body $\Gamma_p K$; and in mechanics it is the center of gravity of a composite mass distribution.

Domain	Centroid object	“Lowest” or extremal meaning
Planar convexity	Barycenter of a convex body	Smallest universal localization or affine-comparison bound
Clustering	Seed or representative point	Lowest-energy initialization or minimum span
Increasing trees	Centroid node	Centroid nearest to the root
$L_p$ convex geometry	$p$-centroid body	Minimal admissible volume
Composite solids	Center of gravity	Minimum height during filling

A persistent misconception is that a centroid must be the Euclidean mean of coordinates. The clustering literature surveyed here explicitly replaces the classical centroid by a coordinate-free pseudo-centroid chosen from the data, while tree theory identifies centroid nodes combinatorially, and convex geometry studies both barycenters of bodies and centroid bodies derived from moment integrals. This suggests that “lowest centroid” is best understood as a family of extremal centroid constructions rather than a single definition.

2. Planar convex bodies: centroid localization and centroid-constrained affine distance

For a planar convex body $A$, Lassak proved that if $H_A$ is any inscribed affine-regular hexagon, then the centroid satisfies

$$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$

and the factor $4/21$ is sharp (Lassak, 2022). In normalized coordinates,

$$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$

the vertical coordinate of the centroid is therefore confined to $[-4/21,\,4/21]$. An explicit pentagon with vertices

$\Gamma_p K$0

has area $\Gamma_p K$1 and centroid $\Gamma_p K$2, showing sharpness.

This localization feeds directly into the centroid-constrained Banach–Mazur theory. For convex bodies $\Gamma_p K$3, the extended Banach–Mazur distance is

$\Gamma_p K$4

while the centroid-constrained variant is

$\Gamma_p K$5

The centroid variant is symmetric,

$\Gamma_p K$6

and, by definition, the unconstrained distance is no larger than the constrained one (Lassak, 2022).

The main planar theorem is

$\Gamma_p K$7

The proof uses Besicovitch’s inscribed affine-regular hexagon, the centroid localization $\Gamma_p K$8, affine normalization to a common regular hexagon, a rotational-scaling map

$\Gamma_p K$9

and a two-stage homothety argument. The relevant extremal factor is

$L_p$0

whose worst case reduces to

$L_p$1

maximized at $L_p$2, yielding $L_p$3 (Lassak, 2022).

The lower-bound side remains open. The paper cites that

$L_p$4

and also gives

$L_p$5

for a regular pentagon $L_p$6 and triangle $L_p$7. The resulting picture is a gap between the universal upper bound $L_p$8 and the known attained value $L_p$9, with the optimal constant in dimension $p$0 still unresolved. A plausible implication is that the sharp centroid localization constant $p$1 is not only a geometric fact about barycenter position, but a structural input into affine comparison problems for non-centrally symmetric bodies.

3. Clustering: lowest-energy initialization and coordinate-free pseudo-centroids

In prototype-based clustering, “lowest centroid” can refer to initialization by minimizing a clustering objective. AQOCI reframes centroid initialization through an NMF-like factorization

$p$2

and minimizes

$p$3

with one-hot constraints

$p$4

The optimization is mapped to QUBO form,

$p$5

equivalent to an Ising Hamiltonian

$p$6

The lowest-energy bitstring encodes the preferred centroid set; AQOCI then adapts scale and offset so that qubit registers represent real-valued centroids rather than only coarse integers (Allgood et al., 2024).

AQOCI’s adaptive step initializes

$p$7

updates scale and offset using a factor $p$8 empirically set to $p$9, and re-anneals iteratively. The evaluated setup used $A$0, $A$1, Gaussian blobs in $A$2D, and a malware dataset reduced to $A$3D by PCA; metrics were inertia, silhouette, homogeneity, completeness, V-measure, and iterations to convergence. The reported outcome was that AQOCI was comparable to classical techniques and superior to QOCI, with nearly identical inertia to random $A$4-means and, on the MOTIF dataset, better silhouette than random initialization (Allgood et al., 2024).

A different extremal notion appears in pseudo-centroid clustering. Here the centroid is always chosen from the cluster itself, so the construction is medoid-like and coordinate-free. For a cluster $A$5 with pairwise distance function $A$6, the MinSum pseudo-centroid is

$A$7

with span

$A$8

and the MinMax pseudo-centroid is

$A$9

with span

$H_A$0

The K-PC framework, together with K-MinSum and K-MinMax, mirrors K-means but uses these separation functions directly on distances that may be non-metric or even negative (Glover, 2016).

The paper also develops diversity-based and intensity-based starting methods, including theorems stating that Primary MinMax and Adaptive MinMax obtain best clusters of the selected admissible size at each construction stage. In this literature, “lowest centroid” means not lowest energy in a Hamiltonian landscape, but the data point minimizing a clusterwise total or worst-case distance. The two clustering traditions therefore share an optimization ethos while using distinct objective functions and admissible centroid classes.

4. Increasing trees: the centroid nearest to the root

For a tree $H_A$1, a node $H_A$2 is a centroid if it minimizes

$H_A$3

or equivalently if none of its branches contains more than half of the other nodes. Every tree has either one or two centroids; two centroids occur only when $H_A$4 is even, they are adjacent, and each has a largest branch of size exactly $H_A$5. In very simple increasing trees, the probability of two centroids vanishes at rate $H_A$6, so the paper adopts the convention that the “lowest centroid” is the centroid nearest to the root (Durant et al., 2018).

Very simple increasing trees include random recursive trees, plane-oriented increasing trees, and $H_A$7-ary increasing trees. Their common parameterization uses

$H_A$8

with $H_A$9 for recursive trees, $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$0 for plane-oriented trees, and $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$1 for $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$2-ary trees. The depth $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$3 of the lowest centroid converges to a discrete limiting variable $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$4 with

$$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$5

In particular, the expected depth tends to $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$6 for plane-oriented trees, $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$7 for recursive trees, and $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$8 for binary increasing trees. The depth is $$g(A)\in T_{c(H_A),\,4/21}(H_A) = c(H_A)+\frac{4}{21}\bigl(H_A-c(H_A)\bigr),$$9, not $4/21$0.

The limiting probability that the root is the centroid is

$4/21$1

with values approximately $4/21$2 for plane-oriented trees, $4/21$3 for recursive trees, and $4/21$4 for binary trees. The centroid label $4/21$5 also converges, with

$4/21$6

The normalized size $4/21$7 of the subtree rooted at the lowest centroid converges to a mixed distribution on $4/21$8 plus an atom at $4/21$9, with density

$$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$0

This notion of “lowest” is combinatorial rather than metric or physical: it refers to minimum depth among centroid nodes. The result is a precise counterpart to the common heuristic that central nodes in recursively grown trees should lie close to the root. Here that heuristic is quantified by explicit limit laws, moments, and root-centroid probabilities.

5. $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$1 centroid bodies and minimal-volume extremizers

In $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$2 convex geometry, the centroid is encoded by the $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$3-centroid body. For a convex body $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$4, the support functions of the $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$5 moment body $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$6 and centroid body $$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$7 are

$$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$8

$$a_1=(1,1),\ a_2=(-1,1),\ a_3=(-2,0),\ a_4=(-1,-1),\ a_5=(1,-1),\ a_6=(2,0),$$9

where

$[-4/21,\,4/21]$0

The $[-4/21,\,4/21]$1 Busemann–Petty centroid inequality states

$[-4/21,\,4/21]$2

equivalently

$[-4/21,\,4/21]$3

with equality if and only if $[-4/21,\,4/21]$4 is a $[-4/21,\,4/21]$5-symmetric ellipsoid centered at the origin (Haddad et al., 2019).

This is a “lowest centroid” statement in the sense of minimal attainable volume. Among origin-centered convex bodies of fixed volume, ellipsoids are exactly the bodies for which the centroid body has the smallest allowed volume, namely the original volume. The affine equivariance

$[-4/21,\,4/21]$6

shows that the extremizers are unique up to linear image, while translation is excluded from the equality characterization because the defining integrals are taken with respect to the origin.

The paper further embeds this in Firey’s $[-4/21,\,4/21]$7 Brunn–Minkowski theory. For $[-4/21,\,4/21]$8, the $[-4/21,\,4/21]$9 mixed volume

$\Gamma_p K$00

satisfies the integral representation

$\Gamma_p K$01

and the $\Gamma_p K$02 Minkowski inequality

$\Gamma_p K$03

The functional extension replaces bodies by functions $\Gamma_p K$04 and $\Gamma_p K$05, defines

$\Gamma_p K$06

and proves a sharp lower bound for

$\Gamma_p K$07

Thus the minimality principle extends from sets to a functional mixed-volume framework (Haddad et al., 2019).

6. Composite solids: the lowest center of gravity during filling

In mechanics, the lowest centroid problem becomes a question about the vertical coordinate of the center of gravity of a container partly filled with a homogeneous material. If $\Gamma_p K$08 is the height of the center of gravity when the fill level is $\Gamma_p K$09, then for the families treated in the paper—cylinders, cones, solids of revolution, power solids, spheres, and half-spheres—the minimum occurs exactly at the unique height $\Gamma_p K$10 satisfying

$\Gamma_p K$11

Equivalently, the center of gravity attains its lowest possible position when it lies on the top surface of the material inside the solid (Sjödin, 2017).

The general moment relations are

$\Gamma_p K$12

$\Gamma_p K$13

where $\Gamma_p K$14 is the cross-sectional area, $\Gamma_p K$15 and $\Gamma_p K$16 are the mass and first moment of the container, and $\Gamma_p K$17 is the fill density. Differentiation gives the unifying differential equation

$\Gamma_p K$18

equivalently

$\Gamma_p K$19

Since $\Gamma_p K$20 while filling, the sign of the derivative is determined by $\Gamma_p K$21: the centroid descends when it lies above the surface and rises when it lies below it. The crossing point is therefore the unique global minimum.

For a cylinder of radius $\Gamma_p K$22 and height $\Gamma_p K$23,

$\Gamma_p K$24

and the extremum equation is

$\Gamma_p K$25

For a top-down right circular cone,

$\Gamma_p K$26

and the extremum condition becomes a cubic. Analogous explicit formulas are derived for spheres and half-spheres. The physical interpretation is entirely Archimedean: the infinitesimal slab added at height $\Gamma_p K$27 contributes a first moment proportional to $\Gamma_p K$28, and the centroid reaches its lowest point exactly when the lever-arm difference vanishes.

The mechanical notion is the most literal reading of “lowest centroid,” but it also clarifies the broader theme. In each of the literatures above, the centroid is singled out by an extremal principle—minimum admissible region, minimum affine distortion, minimum energy, minimum span, minimum depth, minimum volume, or minimum height. The phrase therefore names a class of optimization problems centered on different incarnations of centrality rather than a single universal object.