Liakopoulos’s Volume Estimate

• Liakopoulos’s volume estimate is a pair of distinct frameworks—one in convex geometry and one in real algebraic geometry—that derive volume bounds via structural inequalities.
• The convex-geometric formulation generalizes the dual Loomis–Whitney inequality through a geometric Brascamp–Lieb datum and integral identities with factorial normalization.
• The analytic approach applies a refined global Łojasiewicz inequality combined with tube-volume estimates to bound polynomial sublevel sets and control oscillatory integrals.

Searching arXiv for Liakopoulos and related reverse Brascamp–Lieb work to ground the article in current papers. Liakopoulos's volume estimate denotes two distinct volume-comparison frameworks associated with the name “Liakopoulos” in the supplied literature. In convex geometry, it is the generalized dual Loomis–Whitney inequality for a compact convex set $K\subset \mathbb R^n$ with the origin in its interior, expressed in terms of central sections by linear subspaces and a geometric Brascamp–Lieb datum (Boroczky et al., 17 Jul 2025). In real algebraic geometry and analysis, the same label is used for a scheme that combines a refined global Łojasiewicz inequality with tube-volume bounds for algebraic sets, yielding upper bounds for polynomial sublevel sets and applications to integration indices and oscillatory integrals (Dieu et al., 2017). The two usages are mathematically unrelated in formulation, but both are volume estimates derived from structural inequalities.

1. Convex-geometric formulation

Let $K\subset\mathbb R^n$ be a compact convex set with the origin $o$ in its interior. Let

$$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$

and let $c_1,\dots,c_k>0$ satisfy

$\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$

where $P_{E_i}$ denotes orthogonal projection onto $E_i$. Under this geometric Brascamp–Lieb datum condition, Liakopoulos’s inequality reads

$$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$

Here $|K|$ is the $K\subset\mathbb R^n$0-dimensional volume of the convex body, and $K\subset\mathbb R^n$1 is the $K\subset\mathbb R^n$2-dimensional volume of the central section $K\subset\mathbb R^n$3 (Boroczky et al., 17 Jul 2025).

The geometric meaning of the terms is explicit. The quantity $K\subset\mathbb R^n$4 is the full volume, while $K\subset\mathbb R^n$5 measures the size of the slice of $K\subset\mathbb R^n$6 in the direction $K\subset\mathbb R^n$7. The coefficients $K\subset\mathbb R^n$8 weight the contribution of the sections, and the identity $K\subset\mathbb R^n$9 forces the family $o$0 to be a geometric Brascamp–Lieb datum. Taking traces yields $o$1 (Boroczky et al., 17 Jul 2025).

The classical dual Loomis–Whitney situation appears as a special case: one takes $o$2, so that the sections are coordinate hyperplane sections. In the formulation summarized in the supplied material, this is Meyer’s dual Loomis–Whitney case (Boroczky et al., 17 Jul 2025).

2. Integral structure and factorial normalization

A central feature of the convex-geometric estimate is its reduction to integral identities for the gauge of $o$3. The factorial factors $o$4 and $o$5 arise from

$o$6

where $o$7 is the gauge of $o$8 (Boroczky et al., 17 Jul 2025).

This representation converts a volume inequality into a functional inequality. In particular, one considers the log-concave test function

$o$9

and its restrictions $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$0. A homogeneity argument then yields

$$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$1

which, after inserting the factorial integral identities, becomes exactly Liakopoulos’s volume estimate (Boroczky et al., 17 Jul 2025).

This functional reformulation is significant because it shows that the inequality is not merely combinatorial or section-theoretic; it is an instance of a reverse Brascamp–Lieb mechanism specialized to gauges of convex bodies. A plausible implication is that the structure of equality should be governed by the extremizers of the underlying reverse Brascamp–Lieb inequality rather than by ad hoc convex-geometric arguments alone.

3. Derivation from Barthe’s Geometric Reverse Brascamp–Lieb inequality

The decisive analytic input is Barthe’s Geometric Reverse Brascamp–Lieb inequality. For the same datum $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$2 satisfying $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$3, and for non-negative integrable functions $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$4, the inequality states

$$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$5

where $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$6 denotes the outer integral, needed for measurability issues (Boroczky et al., 17 Jul 2025).

In the deduction of Liakopoulos’s estimate, the strategy is summarized by a sequence of structural steps. One rewrites the relevant volumes as integrals of $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$7; one uses homogeneity to evaluate the sup-integral in Barthe’s inequality at $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$8; and one then inserts the factorial identities to obtain the final volume bound (Boroczky et al., 17 Jul 2025).

The derivation therefore places Liakopoulos’s estimate inside the reverse Brascamp–Lieb formalism. This suggests that the generalized dual Loomis–Whitney inequality is best understood as a convex-body specialization of a broader functional inequality, with the convex geometry encoded by the gauge and the section data encoded by the subspaces $$E_1,\dots,E_k\subset\mathbb R^n \quad\text{be linear subspaces of dimensions }d_i=\dim E_i\in\{1,\dots,n-1\},$$9.

4. Equality characterization in the generalized dual Loomis–Whitney inequality

The supplied material gives an equality characterization via the extremal theory of Barthe’s inequality. The equality case for the Geometric Reverse Brascamp–Lieb inequality is described using a decomposition of $c_1,\dots,c_k>0$0 into “independent subspaces” $c_1,\dots,c_k>0$1 and a “dependent” part $c_1,\dots,c_k>0$2. If each $c_1,\dots,c_k>0$3 and equality holds, then there exist

$c_1,\dots,c_k>0$4

together with parameters $c_1,\dots,c_k>0$5, $c_1,\dots,c_k>0$6, $c_1,\dots,c_k>0$7, a positive-definite map $c_1,\dots,c_k>0$8, and functions $c_1,\dots,c_k>0$9, such that for almost every $\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$0,

$\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$1

Conversely, any collection of $\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$2 of this form is extremal (Boroczky et al., 17 Jul 2025).

When this structure is specialized to $\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$3, the equality analysis simplifies. The auxiliary lemma cited in the supplied material forces all shifts $\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$4 to be zero and makes each $\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$5 log-concave, from which one concludes that

$\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$6

and that the gauge splits as

$\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$7

A further convex-analytic argument shows that if

$\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$8

then $\sum_{i=1}^k c_i\,P_{E_i}\;=\;I_n,$9 for all $P_{E_i}$0, hence $P_{E_i}$1 (Boroczky et al., 17 Jul 2025).

The resulting characterization is that equality in Liakopoulos’s estimate holds if and only if:

1. the independent subspaces $P_{E_i}$2 satisfy $P_{E_i}$3; and
2. the convex body satisfies

$P_{E_i}$4

This is a precise structural description: the extremal body is the convex hull of its lower-dimensional slices along a direct-sum decomposition (Boroczky et al., 17 Jul 2025).

5. Special cases and geometric consequences

Two illustrative specializations are identified in the supplied material. In Meyer’s dual Loomis–Whitney case, one takes $P_{E_i}$5, $P_{E_i}$6, and $P_{E_i}$7. The independent subspaces are then the coordinate axes $P_{E_i}$8, with $P_{E_i}$9, and equality holds precisely when

$E_i$0

Thus the equality bodies are exactly the coordinate cross-polytopal configurations described by those one-dimensional slices (Boroczky et al., 17 Jul 2025).

In the dual Bollobás–Thomason setting, one takes a uniform cover $E_i$1, $E_i$2, and

$E_i$3

The independent subspaces $E_i$4 arise from the induced partition of $E_i$5, and one finds

$E_i$6

This identifies the equality case with a decomposition governed by the partition structure induced by the cover (Boroczky et al., 17 Jul 2025).

These examples clarify a common point of interpretation. The equality theory is not stated merely in terms of arbitrary symmetry or ellipsoidal structure. Instead, it is dictated by direct-sum decompositions into independent subspaces and by reconstruction of the body as the convex hull, or in the Bollobás–Thomason specialization as the corresponding sum, of the relevant slices.

6. Distinct analytic usage: polynomial sublevel-set volume estimates

The supplied material also records a different body of work in which “Liakopoulos’s volume estimate” refers to a method for upper bounds on polynomial sublevel sets. Let $E_i$7 be a real polynomial of total degree $E_i$8, let

$E_i$9

and choose an admissible multi-index $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$0 with

$$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$1

Then there is a constant $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$2, depending only on $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$3, such that for every $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$4 and $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$5,

$$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$6

A weaker but more transparent form is

$$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$7

with $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$8 depending also on $$|K|\;\ge\; \frac{\displaystyle\prod_{i=1}^k d_i!}{n!} \,\prod_{i=1}^k\bigl|\,K\cap E_i\bigr|^{\,c_i}.$$9 and $|K|$0 (Dieu et al., 2017).

The proof combines two ingredients. First, a global Łojasiewicz inequality provides a real algebraic hypersurface $|K|$1 such that

$|K|$2

Second, Wongkew’s tube-volume estimate bounds the volume of $|K|$3-tubes around real algebraic varieties. For a set $|K|$4 of codimension $|K|$5,

$|K|$6

and in the relevant codimension-one case this yields a leading term $|K|$7 (Dieu et al., 2017).

The resulting scheme—described in the supplied material as what one often calls “Liakopoulos’s Volume Estimate”—is the combination of a single-monomial global Łojasiewicz bound with a tube-volume estimate. Its consequences include the lower bound $|K|$8 for the integration index and the oscillatory integral estimate

$|K|$9

for

$K\subset\mathbb R^n$00

(Dieu et al., 2017).

This usage is conceptually separate from the generalized dual Loomis–Whitney inequality. The commonality is only the volume-estimate viewpoint: in one case, volume is bounded below by weighted section volumes of a convex body; in the other, volume of a polynomial sublevel set is bounded above by quantitative control of its tubular localization near an algebraic set.