Symmetric-Maximizer Conjecture

• The Symmetric-Maximizer Conjecture refers to multiple extremal principles suggesting that maximizers are highly symmetric in different mathematical settings
• These conjectures apply to various domains, including convex geometry, spherical varieties, optimization, and Gaussian measures, each with unique symmetry properties
• Mathematical techniques such as variational methods (e.g., hinging) are used to analyze and verify the symmetry of maximizers in specific contexts, illustrating the broad application and significance of this conjecture

The Symmetric-Maximizer Conjecture is not a single conjecture with a uniform mathematical definition, but a name applied to several extremal principles in which a maximizer is conjectured—or proved—to possess a highly symmetric structure. In the literature represented by the cited works, the term concerns isotropic constants of convex bodies, linear programs on spherical data of symmetric varieties, noisy Boolean functions, projective angle energies, Ehrhart roots of symmetric edge polytopes, symmetric tensor matroids, Gaussian concavity and curvature functionals, and permanent optimization over doubly stochastic matrices. The common theme is that an extremal value should be attained by a configuration with maximal available symmetry; however, the relevant functional, admissible class, symmetry group, equality statement, and conjectural status differ substantially from one setting to another.

1. Convex geometry and the isotropic constant

For a $d$-dimensional convex body $K\subseteq\mathbb R^d$, let $X$ be uniformly distributed on $K$, with barycenter $\mu_K=\mathbb E X$ and covariance matrix

$A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$

The isotropic constant is normalized by

$$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$

It is invariant under affine transformations. A body is in isotropic position when $\mu_K=0$ and $A(K)=I_d$; in that position,

$L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$

The associated slicing conjecture asserts that there is a universal constant $K\subseteq\mathbb R^d$0 such that

$K\subseteq\mathbb R^d$1

for every convex body, independently of $K\subseteq\mathbb R^d$2. The best general upper bound cited in the relevant work has the form

$K\subseteq\mathbb R^d$3

whereas a dimension-independent bound remains open. Dimension-independent bounds are known for several classes, including zonoids and bodies possessing an unconditional basis. Maximizers cannot be smooth in a strong sense: a boundary portion that is $K\subseteq\mathbb R^d$4 with positive principal curvatures cannot occur in a maximizing body. These facts motivate the study of nonsmooth polytopes, particularly simplicial polytopes.

The principal theorem for this interpretation is that a simplicial polytope maximizing $K\subseteq\mathbb R^d$5 must be a simplex (Rademacher, 2014). More precisely, if $K\subseteq\mathbb R^d$6 is a $K\subseteq\mathbb R^d$7-dimensional isotropic simplicial polytope that is a local extremum of $K\subseteq\mathbb R^d$8, then for every hyperplane $K\subseteq\mathbb R^d$9 spanned by a $X$0-dimensional face and the origin,

$X$1

where $X$2 is reflection through $X$3. Thus every ridge induces a reflection symmetry. The resulting polytope is isohedral, and all facets are congruent. A theorem of Campi, Colesanti, and Gronchi then implies that the boundary profile associated with such a hyperplane must be affine; simpliciality forces the polytope to be a simplex.

The central variational method is hinging. A facet is rotated around one of its ridges while preserving convexity. Differentiating the isotropic constant under this deformation gives, at an isotropic local extremum, the necessary condition

$X$4

for a weighted distribution $X$5 on the moving slice. For a simplicial facet this yields quadratic equations in the facet vertices. Comparing the equations for two adjacent facets shows that their vertices differ only by the sign of a component orthogonal to the common ridge. The adjacent facets are therefore mirror images.

The result is restricted: it does not prove that every maximizer among arbitrary convex bodies is a simplex, nor that every maximizer has the ridge-reflection symmetries established for simplicial polytopes. The unresolved step is a reduction from general convex bodies—or nonsimplicial and nonsmooth polytopes—to a class admitting this finite facet-based variational analysis.

2. Spherical varieties and the invariant $X$6

A separate use of the name concerns complete spherical varieties. For a complete spherical variety $X$7, one defines a rational invariant

$X$8

where $X$9 is the set of $K$0-invariant prime divisors, $K$1 are valuation vectors, $K$2 are coefficients in Brion’s anticanonical formula

$K$3

and $K$4 is the set of spherically closed spherical roots. The polyhedron $K$5 is

$K$6

The conjecture is

$K$7

with equality if and only if $K$8 is isomorphic to a toric variety (Gagliardi et al., 2014). The term “maximizer” refers to the fact that $K$9 is the supremum of a linear functional over a polyhedral region. In the equality case, the maximizer is characterized geometrically rather than by uniqueness of the optimizing point $\mu_K=\mathbb E X$0.

The invariant has an equivalent linear-programming formulation. If $\mu_K=\mathbb E X$1 is the spherical skeleton of $\mu_K=\mathbb E X$2, then

$\mu_K=\mathbb E X$3

When finite, the dual program gives

$\mu_K=\mathbb E X$4

The conjecture can therefore be expressed combinatorially as

$\mu_K=\mathbb E X$5

Symmetric varieties are spherical varieties whose open orbit is $\mu_K=\mathbb E X$6, with

$\mu_K=\mathbb E X$7

for a nontrivial involution $\mu_K=\mathbb E X$8 of $\mu_K=\mathbb E X$9. Their spherical systems decompose into diagonal embeddings and quotients by spherically closed symmetric subgroups. This classification permits a reduction to complete reduced elementary symmetric skeletons.

For a skeleton $A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$0, elementary replacement and reduction produce skeletons $A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$1 and $A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$2 satisfying

$A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$3

A monotonicity argument then reduces the estimate to skeletons with one marked valuation. Explicit primal and dual linear-programming computations for all symmetric Luna diagrams establish

$A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$4

for every complete symmetric skeleton. Equality occurs exactly for linear spherical skeletons arising from multiplicity-free representations, which correspond through the Cox-ring argument to toric varieties.

Thus the conjecture is proved for symmetric varieties, while it remains conjectural for arbitrary complete spherical varieties. It is also stronger than the generalized Mukai inequality in the spherical setting: the conjecture implies

$A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$5

for suitable smooth Fano varieties, with the equality case yielding

$A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$6

It also yields a combinatorial smoothness criterion through equality for localized spherical skeletons.

3. Symmetric extremizers in discrete and algebraic optimization

Several conjectures with this name concern finite-dimensional optimization in which the proposed optimizer is the most symmetric available object.

Hypergraph Lagrangians

For an $A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$7-uniform hypergraph $A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$8, define

$A(K)=\mathbb E\bigl[(X-\mu_K)(X-\mu_K)^T\bigr].$9

If $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$0, the principal Frankl–Füredi conjecture asserts

$$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$1

with equality if and only if $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$2 is an integer. In the equality case, the extremal hypergraph is the complete $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$3-graph on $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$4 vertices and the maximizing vector is uniform on those vertices (Nikiforov, 2018).

The theorem is proved for every $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$5 and for every $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$6 satisfying

$$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$7

The argument uses Lagrange multiplier conditions, support reduction, and estimates for elementary symmetric functions. If $$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$8 is an eigenvector of the hypergraph, then

$$L_K^{2d}=\frac{\det(A(K))}{\operatorname{vol}(K)^2}.$$9

for every positive coordinate. A key estimate is

$\mu_K=0$0

Bounds for $\mu_K=0$1 then force equality only when the positive coordinates are equal.

The result establishes the symmetric-maximizer prediction only in the stated ranges. The remaining cases with $\mu_K=0$2 and relatively small $\mu_K=0$3 are not resolved by this argument, and the paper does not prove the full Frankl–Füredi conjecture.

Permanent optimization

For the Birkhoff polytope

$\mu_K=0$4

consider

$\mu_K=0$5

For $\mu_K=0$6, Kim and Roush conjectured

$\mu_K=0$7

This is proved with complete equality classification in (Lavi, 9 Aug 2026). The unique maximizers up to simultaneous permutation conjugation are

$\mu_K=0$8

The first block is a symmetric weighted triangle, and each remaining block is a transposition. Since

$\mu_K=0$9

one obtains

$A(K)=I_d$0

The proof expands the permanent through random functional digraphs: $A(K)=I_d$1 The dominant layer consists of $A(K)=I_d$2 transpositions and one nonperiodic vertex. A triangle-charge inequality,

$A(K)=I_d$3

controls the contribution of the top even-cycle layer relative to maps with one directed triangle and $A(K)=I_d$4 transpositions. Equality forces the support digraph to split into one triangle block and transposition blocks. The theorem is a genuine emergent-symmetry result: symmetry is not imposed on $A(K)=I_d$5, but every maximizer is symmetric after simultaneous relabeling.

Symmetric tensor matroids

For a generic realization $A(K)=I_d$6, the symmetric tensor matroid $A(K)=I_d$7 assigns to an edge $A(K)=I_d$8 the tensor

$A(K)=I_d$9

The symmetric-maximizer conjecture in this setting asserts that the generic symmetric tensor matroid $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$0 is the unique maximal abstract symmetric $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$1-tensor matroid. By matroid duality, this is equivalent to Graver’s maximality conjecture for generic rigidity matroids (Jackson et al., 18 Mar 2025).

The conjecture is proved for

$L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$2

The proof characterizes $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$3-independence through a recursively defined family of cyclic flats $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$4 and establishes

$L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$5

The additional complete-bipartite circuits

$L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$6

are essential. If only the star circuits $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$7 are retained, the resulting family has multiple maximal elements for $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$8. The full conjecture remains open outside the range $L_K^{2d}=\frac1{\operatorname{vol}(K)^2}.$9; a possible obstruction appears at $K\subseteq\mathbb R^d$00.

4. Boolean, projective, and Ehrhart formulations

Noisy Boolean functions

Let $K\subseteq\mathbb R^d$01 be balanced, and let $K\subseteq\mathbb R^d$02 be the binary noise operator for $K\subseteq\mathbb R^d$03. The Li–Médard conjecture asserts that dictatorships maximize

$K\subseteq\mathbb R^d$04

for $K\subseteq\mathbb R^d$05. A symmetrized version uses

$K\subseteq\mathbb R^d$06

The dictatorship output takes only the values $K\subseteq\mathbb R^d$07 and $K\subseteq\mathbb R^d$08, and therefore

$K\subseteq\mathbb R^d$09

The balanced Courtade–Kumar conjecture states

$K\subseteq\mathbb R^d$10

where $K\subseteq\mathbb R^d$11 is binary entropy. Its entropy expression is symmetric in $K\subseteq\mathbb R^d$12 and $K\subseteq\mathbb R^d$13. The paper proves that the balanced Courtade–Kumar conjecture is equivalent to the symmetrized Li–Médard conjecture, while an unsymmetrized Courtade–Kumar formulation is equivalent to the unsymmetrized Li–Médard conjecture (Barnes et al., 2020).

The bridge is differentiation at $K\subseteq\mathbb R^d$14: $K\subseteq\mathbb R^d$15 The converse implication uses Laguerre’s zero-counting theorem for exponential sums. Differences such as

$K\subseteq\mathbb R^d$16

are exponential polynomials in $K\subseteq\mathbb R^d$17, so their zero structure is constrained by sign changes in their coefficients. Fourier analysis supplies the endpoint $K\subseteq\mathbb R^d$18, where dictatorships are optimal because the largest nonconstant noise eigenvalue occurs at Fourier weight one.

The result is an equivalence theorem, not a proof of dictatorship optimality. The unresolved intermediate noise regime remains open.

Projective angle energies

For unoriented lines in $K\subseteq\mathbb R^d$19, identify antipodal points on $K\subseteq\mathbb R^d$20 and define the normalized angle kernel

$K\subseteq\mathbb R^d$21

For a probability measure $K\subseteq\mathbb R^d$22 on $K\subseteq\mathbb R^d$23,

$K\subseteq\mathbb R^d$24

The Fejes Tóth conjecture asserts that $K\subseteq\mathbb R^d$25 is maximized by distributing mass as evenly as possible among the $K\subseteq\mathbb R^d$26 axes of an orthonormal basis. The paper introduces the entire family $K\subseteq\mathbb R^d$27, $K\subseteq\mathbb R^d$28, and proves that the original conjecture is equivalent to the same coordinate-axis configuration being the optimizer for every finite $K\subseteq\mathbb R^d$29 (Lim et al., 2020).

The limiting case $K\subseteq\mathbb R^d$30 is completely solved. Here

$K\subseteq\mathbb R^d$31

so the energy counts orthogonal pairs. The essentially unique optimizer is the uniform measure on an orthonormal basis: $K\subseteq\mathbb R^d$32 For fixed $K\subseteq\mathbb R^d$33, the points are distributed with multiplicities $K\subseteq\mathbb R^d$34 on $K\subseteq\mathbb R^d$35 axes and $K\subseteq\mathbb R^d$36 on the remaining axes. The original $K\subseteq\mathbb R^d$37 conjecture remains open for $$d\geq2.</p> <h3 class='paper-heading' id='symmetric-edge-polytopes'>Symmetric edge polytopes</h3> <p>For a complete multipartite graph$$K\subseteq\mathbb R^d$38, the symmetric edge polytope is

$K\subseteq\mathbb R^d$39

Its Ehrhart polynomial satisfies a reciprocity relation forcing roots to be symmetric about

$K\subseteq\mathbb R^d$40

The conjecture is that all roots lie on $K\subseteq\mathbb R^d$41 and that certain neighboring multipartite Ehrhart polynomials interlace there (Kölbl, 2024).

The conjecture is proved for several families, including

$K\subseteq\mathbb R^d$42

and additional interlacing relations are established conditionally. The methods combine explicit $K\subseteq\mathbb R^d$43-polynomials, Gröbner bases, unimodular triangulations, directed spanning-tree enumeration, and positive-coefficient interlacing recursions. The approach does not settle the general conjecture: higher cross-degree introduces too many auxiliary interlacings, recursion coefficients can become negative, and graphs containing $K\subseteq\mathbb R^d$44 require cubic Gröbner relations.

5. Gaussian symmetric extremizers

A further interpretation concerns the strongest concavity inequality for Gaussian measure under Minkowski interpolation of centrally symmetric convex sets. For standard Gaussian measure $K\subseteq\mathbb R^d$45, the desired inequality has the form

$K\subseteq\mathbb R^d$46

The conjectured optimal function is determined by round cylinders

$K\subseteq\mathbb R^d$47

where $K\subseteq\mathbb R^d$48 is chosen so that $K\subseteq\mathbb R^d$49. Writing

$K\subseteq\mathbb R^d$50

and $K\subseteq\mathbb R^d$51, one has

$K\subseteq\mathbb R^d$52

The cylinder concavity power is

$K\subseteq\mathbb R^d$53

where

$K\subseteq\mathbb R^d$54

The conjecture asserts that every symmetric convex $K\subseteq\mathbb R^d$55 with $K\subseteq\mathbb R^d$56 satisfies

$K\subseteq\mathbb R^d$57

with equality only for an appropriate round cylinder (Livshyts, 2021).

The paper does not prove this sharp comparison. It proves instead a torsional-rigidity lower bound: $K\subseteq\mathbb R^d$58 with equality precisely for round cylinders. The proof uses a Gaussian Hessian inequality, Brascamp–Lieb equality cases, and a stability theorem. A weaker, non-sharp $K\subseteq\mathbb R^d$59-concavity inequality is also established.

The same cylinders arise in the symmetric Gaussian isoperimetric problem, but the problems are distinct. Gaussian concavity concerns the largest admissible concavity exponent, whereas Gaussian isoperimetry concerns minimizing Gaussian perimeter. In a related curvature problem, the Gaussian-weighted total mean curvature

$K\subseteq\mathbb R^d$60

is maximized by centered disks among planar convex sets of fixed Gaussian measure, but the analogous unrestricted assertion fails in higher dimensions (Fusco et al., 2023). Long symmetric cylinders exceed the ball for sufficiently small Gaussian mass in dimension three. For small centrally symmetric perturbations, the ball is even a local minimizer when

$K\subseteq\mathbb R^d$61

For sufficiently large radius,

$K\subseteq\mathbb R^d$62

the ball is a local maximizer against nonsymmetric perturbations. A global maximality theorem is available under a uniform curvature bound and sufficiently large Gaussian mass, but the unrestricted higher-dimensional problem remains open.

6. Counterexamples, scope, and terminology

The various results show that “symmetric maximizer” has no universal logical content beyond an extremal claim involving symmetry. Symmetry may be:

• A conjectured structure of extremizers, as for simplicial maximizers of the isotropic constant, Fejes Tóth angle energies, and symmetric Gaussian concavity.
• A theorem in a restricted parameter range, as for hypergraph Lagrangians, symmetric varieties, Ehrhart roots, and symmetric tensor matroids.
• An emergent equality property, as for the permanent of $K\subseteq\mathbb R^d$63 over doubly stochastic matrices, where maximizers are forced to be symmetric after simultaneous relabeling.
• A false global principle, as demonstrated by the Lyapunov-operator counterexample.

For the Lyapunov operator

$K\subseteq\mathbb R^d$64

with Frobenius norm, the symmetric-maximizer conjecture asserted

$K\subseteq\mathbb R^d$65

where $K\subseteq\mathbb R^d$66 and $K\subseteq\mathbb R^d$67 are the symmetric and skew-symmetric matrix subspaces. The conjecture holds for $K\subseteq\mathbb R^d$68, remains open for $K\subseteq\mathbb R^d$69, and is false for every $K\subseteq\mathbb R^d$70 (Kressner et al., 21 Aug 2026).

An explicit integer matrix of order seven satisfies

$K\subseteq\mathbb R^d$71

The strict inequalities are certified exactly: a rational positive-definite $K\subseteq\mathbb R^d$72 factorization proves the symmetric upper bound, while an explicit integer skew-symmetric test matrix proves the lower bound. Direct sums with zero blocks extend the counterexample to all $K\subseteq\mathbb R^d$73.

This counterexample illustrates an important distinction. In several successful settings, symmetry is forced by the geometry or combinatorics of the extremal problem. But permutation invariance of an admissible class, convexity, or decomposition into symmetric and antisymmetric subspaces does not by itself imply that a maximizer lies in the symmetric component. The Lyapunov example shows that skew-symmetric structure can dominate even when the full operator preserves both subspaces.

Taken collectively, the results support a conditional methodological principle: extremal problems often become tractable when a variational, algebraic, or combinatorial argument forces the optimizer into a highly structured symmetric class. They also delimit that principle. Some conjectures are established only for special families or parameter ranges; some are equivalent to other difficult conjectures; some admit exact limiting-case theorems; and others fail beyond a critical dimension. The phrase Symmetric-Maximizer Conjecture therefore denotes a family of structurally related but mathematically distinct conjectures rather than one universal statement.