Weak Isomorphic Reverse Isoperimetry Conjecture

• The weak isomorphic reverse isoperimetry conjecture posits that every origin-symmetric convex body, after a volume-preserving linear transformation, contains a large inner body with an isoperimetric quotient of order √n.
• It connects geometric optimization with applications in stochastic clustering, Lipschitz extension, and affine-invariant shape analysis, with confirmed cases for convex polytopes having O(n) facets.
• Partial results link metric separation, spectral estimates, and volumetric inequalities to the existence of nearly Euclidean inner models, showcasing an isomorphic relaxation of classical reverse isoperimetric bounds.

Searching arXiv for recent and directly relevant papers on the weak isomorphic reverse isoperimetry conjecture and adjacent reverse-isoperimetric formulations. Weak isomorphic reverse isoperimetry is a conjectural strengthening of reverse isoperimetric theory in asymptotic convex geometry. For an origin-symmetric convex body $K\subset \mathbb R^n$, it asks whether, after a volume-preserving linear change of coordinates, one can find a large-volume origin-symmetric inner body whose isoperimetric quotient is of Euclidean order $\sqrt n$. In its explicit form, the conjecture was formulated in connection with stochastic clustering, Lipschitz extension, and affine-invariant shape optimization, and it was later verified for convex polytopes with $O(n)$ facets (Naor, 2021, Ball et al., 17 Sep 2025).

1. Formal statement and basic variants

For a convex body $K\subset \mathbb R^n$, the isoperimetric quotient is

$$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$

It is scale-invariant, and the Euclidean ball satisfies

$iq(B_{\ell_2^n})\asymp \sqrt n.$

The strong isomorphic reverse isoperimetry conjecture asserts the existence of a universal constant $c>0$ such that for every origin-symmetric convex body $K\subset \mathbb R^n$, there exist $S\in SL_n(\mathbb R)$ and an origin-symmetric convex body $L\subset \mathbb R^n$ with

$\sqrt n$0

and

$\sqrt n$1

The weak version relaxes the lower inclusion. It asks whether for every origin-symmetric convex body $\sqrt n$2 there exist $\sqrt n$3 and an origin-symmetric convex body $\sqrt n$4 such that

$\sqrt n$5

Thus the weak form retains only two requirements: $\sqrt n$6 must lie inside a volume-preserving image of $\sqrt n$7, and its volume radius must remain comparable to that of $\sqrt n$8 (Naor, 2021).

A symmetric variant is formulated for normed spaces with enough symmetries whose isometry group is a subgroup of $\sqrt n$9. In that setting one asks for a normed space $O(n)$0 with

$O(n)$1

and

$O(n)$2

This is the version most directly tied to the metric applications developed in the literature (Naor, 2021).

2. Place within reverse isoperimetric theory

The classical Euclidean isoperimetric inequality gives a lower bound on $O(n)$3, with the Euclidean ball as minimizer. Reverse isoperimetry asks for upper bounds after a suitable normalization. In the symmetric setting, Ball’s reverse isoperimetric theorem gives, for every origin-symmetric convex body $O(n)$4, a volume-preserving linear map $O(n)$5 such that

$O(n)$6

For general convex bodies, the affine invariant

$O(n)$7

satisfies

$O(n)$8

These bounds are of order $O(n)$9, not $K\subset \mathbb R^n$0 (Naor, 2021, Ball et al., 17 Sep 2025).

The weak isomorphic conjecture asks whether the affine upper bound of order $K\subset \mathbb R^n$1 can be reduced to Euclidean order $K\subset \mathbb R^n$2 once one is allowed to pass from $K\subset \mathbb R^n$3 to a large-volume inner body $K\subset \mathbb R^n$4. This is the precise sense in which the conjecture is both reverse and isomorphic. It is reverse because it seeks an upper bound on $K\subset \mathbb R^n$5, contrary to the classical lower bound; it is isomorphic because it does not require the original body, or even its affine image, to have Euclidean-order isoperimetry, only a quantitatively large inner model.

A common misconception is that the conjecture predicts $K\subset \mathbb R^n$6 for the affine image itself. The weak form does not say this. The improvement is obtained by passing to an inner body $K\subset \mathbb R^n$7, and the large-volume condition is formulated only at the level of volume radius.

3. Metric and functional-analytic reformulations

The conjecture was introduced in a framework where volumetric inequalities control metric partition and extension phenomena. The key metric quantity is the separation modulus $K\subset \mathbb R^n$8 of a metric space $K\subset \mathbb R^n$9, defined as the infimum of $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$0 such that for every $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$1 there exists a $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$2-separating $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$3-bounded random partition $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$4 satisfying

$$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$5

For normed spaces, the paper relates $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$6 to volumetric invariants and projection bodies. If $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$7, then

$$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$8

Here $$iq(K)=\frac{\operatorname{vol}_{n-1}(\partial K)}{\operatorname{vol}_n(K)^{\frac{n-1}{n}}}.$$9 is the external volume ratio, $iq(B_{\ell_2^n})\asymp \sqrt n.$0 is the dual normed space, and $iq(B_{\ell_2^n})\asymp \sqrt n.$1 is the projection body associated with $iq(B_{\ell_2^n})\asymp \sqrt n.$2 (Naor, 2021).

This upper bound is the direct motivation for weak isomorphic reverse isoperimetry. The problem becomes an optimization over inner bodies $iq(B_{\ell_2^n})\asymp \sqrt n.$3: one wants $iq(B_{\ell_2^n})\asymp \sqrt n.$4 large in volume and simultaneously favorable for the projection-body functional. Proposition 1.12 of the same paper identifies this optimization with a weak reverse isoperimetric statement: large volume radius together with $iq(B_{\ell_2^n})\asymp \sqrt n.$5 is equivalent, up to constants, to controlling the relevant projection functional by $iq(B_{\ell_2^n})\asymp \sqrt n.$6 (Naor, 2021).

The same program interacts with Lipschitz extension. Lee–Naor’s inequality

$iq(B_{\ell_2^n})\asymp \sqrt n.$7

links the Lipschitz extension modulus to stochastic separation. Consequently, if the weak conjecture were true in the symmetric settings under consideration, then the upper bound on $iq(B_{\ell_2^n})\asymp \sqrt n.$8 would match the lower bound up to constants, yielding

$iq(B_{\ell_2^n})\asymp \sqrt n.$9

for spaces with enough symmetries. This is the mechanism by which reverse isoperimetry becomes a tool in the geometry of finite-dimensional normed spaces (Naor, 2021).

4. Partial results and approximate forms before the polytope theorem

Several nontrivial instances and approximations were established before the polytope case was isolated. For $c>0$0, the strong isomorphic reverse isoperimetry conjecture is proved: there exists a norm $c>0$1 with $c>0$2 and

$c>0$3

The conjectural picture is also stable under unconditional composition: if suitable weak reverse isoperimetric statements hold for component spaces, then they persist, up to constants, for unconditional direct sums built from them (Naor, 2021).

For arbitrary symmetric convex bodies, an approximate weak form is known through intersections with Euclidean balls. If

$c>0$4

for a suitable $c>0$5, then

$c>0$6

where $c>0$7 is the $c>0$8-convexity constant. Using Pisier’s estimate $c>0$9, this yields an $K\subset \mathbb R^n$0-loss version of the weak conjecture. One consequence is that for canonically positioned spaces,

$K\subset \mathbb R^n$1

This does not establish the conjectured $K\subset \mathbb R^n$2-scale exactly, but it shows that the correct exponent is already visible in full generality (Naor, 2021).

The same paper derives concrete metric consequences from these approximate reverse-isoperimetric inputs. Among them are

$K\subset \mathbb R^n$3

and

$K\subset \mathbb R^n$4

These are not proofs of the weak conjecture itself, but they show that the conjecture sits inside a wider program in which Euclidean-order isoperimetry is expected to govern extension and partition behavior (Naor, 2021).

5. Convex polytopes and the $K\subset \mathbb R^n$5-facet theorem

A decisive recent advance concerns convex polytopes with few facets. If $K\subset \mathbb R^n$6 is a convex polytope with $K\subset \mathbb R^n$7 facets, then there exist a vector $K\subset \mathbb R^n$8, a matrix $K\subset \mathbb R^n$9, and an origin-symmetric convex body $S\in SL_n(\mathbb R)$0 such that

$S\in SL_n(\mathbb R)$1

In particular, if $S\in SL_n(\mathbb R)$2, then

$S\in SL_n(\mathbb R)$3

This proves the weak isomorphic reverse isoperimetry conjecture for $S\in SL_n(\mathbb R)$4-dimensional convex polytopes with $S\in SL_n(\mathbb R)$5 facets (Ball et al., 17 Sep 2025).

When $S\in SL_n(\mathbb R)$6 is origin-symmetric, the translation can be removed. The paper observes that

$S\in SL_n(\mathbb R)$7

so in the symmetric case one genuinely obtains $S\in SL_n(\mathbb R)$8, matching the original formulation of the conjecture (Ball et al., 17 Sep 2025).

The proof proceeds through a spectral reformulation. For a convex body $S\in SL_n(\mathbb R)$9, let $L\subset \mathbb R^n$0 be the first Dirichlet eigenvalue of $L\subset \mathbb R^n$1 on $L\subset \mathbb R^n$2. The paper proves that a polytope with $L\subset \mathbb R^n$3 facets admits a positive definite linear image $L\subset \mathbb R^n$4 satisfying

$L\subset \mathbb R^n$5

After normalization to determinant $L\subset \mathbb R^n$6, this estimate is combined with the Cheeger body $L\subset \mathbb R^n$7, whose convexity and uniqueness are used to extract an inner body with the claimed volume and isoperimetric properties. The argument uses the polyhedral representation of $L\subset \mathbb R^n$8 as an intersection of slabs, a Brascamp–Lieb-type factorization of facet normals, and the spectral-to-isoperimetric reduction developed in earlier work (Ball et al., 17 Sep 2025).

The same paper places this theorem against a sharp approximate-isoperimetric background. If $L\subset \mathbb R^n$9 denotes the smallest possible isoperimetric quotient of an $\sqrt n$00-dimensional polytope with $\sqrt n$01 facets, then

$\sqrt n$02

For origin-symmetric polytopes with $\sqrt n$03 vertices, one also has the sharp affine reverse bound

$\sqrt n$04

These results show that the weak conjecture is not a formal consequence of ordinary affine reverse isoperimetry: even after affine normalization, exact Euclidean-order control of the original polytope is generally unavailable, while the inner-body formulation can succeed (Ball et al., 17 Sep 2025).

6. Status, scope, and points of interpretation

The conjecture remains open in full generality. The $\sqrt n$05-facet theorem is a partial confirmation, not a complete resolution. For polytopes with $\sqrt n$06, the available bound is

$\sqrt n$07

which may be much larger than $\sqrt n$08. For arbitrary convex bodies, only approximate forms with logarithmic losses are presently established (Naor, 2021, Ball et al., 17 Sep 2025).

Several distinctions are essential.

First, the weak conjecture is strictly weaker than the strong isomorphic reverse isoperimetry conjecture. The strong version requires two-sided control

$\sqrt n$09

whereas the weak version keeps only the inner inclusion and comparable volume radius.

Second, the weak conjecture does not assert that $\sqrt n$10 itself has Euclidean-order isoperimetric quotient. The affine image may still have $\sqrt n$11; the point is the existence of an inner body $\sqrt n$12 with $\sqrt n$13.

Third, the conjecture is not merely an abstract reformulation of Ball’s theorem. Ball’s theorem gives order $\sqrt n$14, which is sharp for exact affine reverse isoperimetry in several polyhedral families. The weak conjecture seeks an order-$\sqrt n$15 bound only after an isomorphic relaxation.

A plausible implication is that the conjecture should be viewed as a structural statement about the existence of a large nearly Euclidean isoperimetric core inside every volume-preserving affine image of a symmetric convex body, rather than as a claim that the ambient body itself becomes nearly Euclidean.

7. Related reverse-isoperimetric and reverse-isodiametric phenomena

The phrase “reverse isoperimetry” occurs in several mathematically distinct settings, and these help delimit the scope of the weak isomorphic conjecture.

In symplectic and almost complex geometry, a semi-local reverse isoperimetric inequality controls the boundary length of a $\sqrt n$16-holomorphic curve by the area of the portion of the curve lying near a totally real boundary condition: $\sqrt n$17 This is a local analytic monotonicity statement rather than a convex-geometric isomorphic problem (Duval, 2015).

In Gaussian space, reverse isoperimetry appears as a perimeter-maximization problem for convex sets under Gaussian measure. For the classes $\sqrt n$18 and $\sqrt n$19 studied in dimension $\sqrt n$20, the Gaussian perimeter satisfies

$\sqrt n$21

and extremal behavior is realized only by degenerating sequences collapsing to a line. The paper explicitly describes these as weak or isomorphic reverse isoperimetric inequalities on restricted subclasses (Brock et al., 27 Mar 2025).

There is also a close but distinct reverse-isodiametric line of work. For convex bodies $\sqrt n$22, the isodiametric quotient

$\sqrt n$23

has no dimension-only lower bound, but after linear renorming one can seek universal reverse bounds. In the $\sqrt n$24-symmetric case, the sharp inequality

$\sqrt n$25

is proved, with equality exactly for regular crosspolytopes; in the general case, strong asymptotic lower bounds are obtained via Behrend position, Löwner position, and Dvoretzky–Rogers-type simplex estimates (Merino et al., 2018). In dimension $\sqrt n$26, the exact Makai Jr. constant is established: $\sqrt n$27 with simplices as extremals (Aliev, 2023).

These neighboring theories do not prove the weak isomorphic reverse isoperimetry conjecture, but they indicate a recurrent pattern. Exact reverse inequalities often require linear renorming, restriction to specific geometric classes, passage to inner models, or acceptance of degenerate extremizing behavior. This suggests that the weak conjecture belongs to a broader family of reverse geometric principles in which Euclidean-order bounds emerge only after a controlled structural relaxation.