Milman's reverse Brunn–Minkowski inequality¶

Two centrally symmetric convex bodies admit a volume-preserving relative position in which their Minkowski sum has a dimension-free upper bound on its volume radius.

Version: v1 · 2026-10-07 · History
Domain-specific #: 13945
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Convex Geometry, Geometric Functional Analysis → Mathematics
Aliases: Reverse Brunn Minkowski Inequality

Core Idea¶

Milman's reverse Brunn–Minkowski inequality gives an upper bound on the size of a Minkowski sum after its two convex bodies are suitably positioned relative to each other. For centrally symmetric, full-dimensional compact convex bodies (K,L\subset\mathbb R^n), write (|K|) for volume and (K+L={x+y:x\in K,y\in L}). The unit-coefficient form says there is a real linear map (T) with (\det T=1) and a numerical (C), independent of (n,K,L), such that

\[ |K+T(L)|^{1/n}\le C\bigl(|K|^{1/n}+|L|^{1/n}\bigr). \]

The (1/n) powers are volume radii: they put all three volumes on a length scale. The map changes relative shape or orientation while preserving (|L|), so the two sides remain comparable. This is a theorem about existence of a favorable relative position, not an upper bound valid for every original placement.^[1]

The ordinary Brunn–Minkowski inequality points the other way: it lower-bounds the volume radius of (K+L) by the sum of the input volume radii without searching for a new position. The reverse theorem does not undo that lower bound. It adds a dimension-free upper comparison once a permitted map has been chosen. Both direction and positioning condition belong in the name's meaning.^[1]

Structural Signature¶

Admissible bodies and dimension. The cited original theorem takes centrally symmetric compact convex bodies in a common (\mathbb R^n), with positive volume for the volume-radius comparison. A different body class requires a separately sourced extension.^[1]
Relative volume-preserving map. A map (T) with (\det T=1) acts on one body. Its role is existential: for a favorable pair, (T=I) may already work. The theorem does not say every determinant-one map works.^[1]
Minkowski addition. The output body is (K+T(L)), all pairwise vector sums. Replacing this with an intersection, union or polar-body product changes the statement.^[1]
Root-volume upper comparison. The (n)-th root of the positioned sum's volume is bounded by (C) times the sum of input root-volumes. Rooting is what allows one constant across dimensions.^[1]
Uniformity. (C) does not depend on (n), body shapes or how elongated an input may be. Exact constants for selected pairs are illustrative and cannot establish the worst-case universal value.^[1]

What It Is Not¶

An upper bound at every fixed relative orientation is false. Two thin ellipses with long axes at right angles each keep area (\pi), yet their unpositioned sum contains an increasingly large square. Their explicit repair below shows exactly what the determinant-one freedom accomplishes. A general upper estimate whose factor grows with eccentricity or dimension is also weaker than Milman's result.^[1]

A pair already well placed does not prove that all pairs require no correction. The high-dimensional cube/cross-polytope example below satisfies a dimension-free estimate with (T=I) merely because one body lies inside the other. That is an easy instance, not a proof that identity is always a technical M-position. The original preprint's polar-body companion and normed-space application are related results, not the two worked body-pair examples here; reverse Santaló is a different inequality.^[1]

Scope of Application¶

The theorem belongs to convex geometry and finite-dimensional geometric functional analysis. Its original statement covers centrally symmetric compact convex bodies and gives a constant uniform in ambient dimension. Normed-space unit balls are symmetric convex bodies, which explains why Milman develops the result alongside polar bodies and a Euclidean-structure corollary. Those applications should be named as consequences or contexts, not substituted for concrete cases of the same (K+T(L)) bound.^[1]

The two examples here are elementary calculations derived for this entry from the theorem's hypotheses and Minkowski addition. They were independently checked but are not examples printed in Milman's preprint. The planar ellipse pair shows the need for a relative adjustment; the cube/cross-polytope family supplies an unlike, polyhedral pair in every dimension (n\ge2) where identity already suffices. Neither calculation proves the universal theorem.^[1]

Clarity¶

Always state whether the bodies are in their initial position or after (T). The determinant-one condition preserves one summand's volume; it does not preserve the Minkowski-sum volume. The theorem chooses some such (T). The formula above uses unit coefficients and only this form is needed here; no formula for arbitrary weights is inferred from the displayed examples.^[1]

Separate three constants: an exact ratio for one body pair, a bound uniform within a displayed family, and the universal (C) over all admissible bodies and dimensions. The aligned ellipses have ratio (1); the cube family has ratio at most (2). Those figures are not a claim about the optimal universal (C), and the identity map for the cube pair is not asserted to construct a technical M-position.

Manages Complexity¶

Without a position choice, bodies can have the same volume but incompatible directions of elongation. Their sum may fill far more space than either body. Milman's theorem separates size from position: compare sizes through volume radii, then allow a volume-preserving map to control the relative geometry. A universal constant controls the resulting sum across dimensions, even though the favorable map depends on the bodies.^[1]

This compression does not prescribe a simple map for an arbitrary pair. The ellipse map can be written explicitly because the axes are known; the theorem establishes existence more generally. For a pair already nested as in the cube example, a direct inclusion gives a bound without using the full theorem. These are distinct levels of argument.

Abstract Reasoning¶

Take two admissible bodies and compute or bound their individual root-volumes. If a proposed reverse inequality is asserted in the current position, test anisotropic counterexamples first. For the perpendicular ellipses, a square sits in the sum and forces the ratio to grow with eccentricity. Then ask whether a determinant-one relative map changes the alignment while keeping each body's volume fixed. In the worked pair, it does so exactly.^[1]

In another pair, first look for an inclusion that already bounds (K+L). When (L\subset K), monotonicity of Minkowski addition gives (K+L\subset K+K=2K); the cube/cross-polytope case converts that to a dimension-free root-volume estimate with (T=I). The theorem's force is that some bounded position exists even when no such easy inclusion is available.^[1]

Knowledge Transfer¶

The portable lesson within the mathematical setting is that a failed reverse comparison at fixed position can become a uniform comparison after an invariant-preserving reconfiguration. Here the invariant is volume, the reconfiguration is a determinant-one linear map, and the target comparison is a Minkowski-sum volume radius. The exact convex-body machinery cannot be dropped: an arbitrary “normalize and compare” metaphor does not instantiate Milman's theorem.^[1]

The live Formal Theorem parent captures this as a proved mathematical statement; the live Convex Body parent captures its necessary operands. Formal theorems need not mention geometry, and convex bodies need not appear in this inequality. Their inherited paths to Formal System and Convexity supply no reason to duplicate those Primes as direct edges.

Examples¶

Canonical: perpendicular planar ellipses¶

Let (a>1). Take (K={(x,y):x^2/a2+a^2y2\le1}) and (L={(x,y):a^2x2+y^2/a2\le1}). Both are centrally symmetric ellipses of area (\pi). The map (T=\operatorname{diag}(a^2,a)) has determinant (1) and sends (L) to (K). Hence (K+T(L)=2K), its area is (4\pi), and its area radius is (2\sqrt\pi=|K|^{½}+|L|). The reverse ratio after positioning is exactly (1).^[1]

Before this map, (K) contains the horizontal segment ([-a,a]\times{0}) and (L) the vertical segment ({0}\times[-a,a]). Their sum therefore contains ([-a,a]^2), of area (4a^2). The unpositioned area radius is at least (2a), while the input radius sum stays (2\sqrt\pi). Its ratio grows at least as (a/\sqrt\pi). Mapped back: same-volume convex inputs, bad relative axes, volume-preserving correction, controlled sum. These computations are derived here; Milman's source supplies the general theorem, not this printed ellipse example.

Applied contrast: cube and cross-polytope¶

For each integer (n\ge2), take (K=[-1,1]^n) and (L={x\in\mathbb R^n:\sum_i|x_i|\le1}). The second body is the cross-polytope and lies inside the cube. Their volumes are (2^n) and (2^n/n!). With (T=I), inclusion gives (K+L\subset K+K=2K), hence (|K+L|^{1/n}\le4). The input root-volumes sum to (2+2/(n!)^{1/n}\ge2), so this pair's reverse ratio is at most (2) in every such dimension.^[1]

Mapped back: symmetric convex polyhedral inputs, determinant-one identity position, Minkowski sum, dimension-free upper ratio. This pair is already favorable by nesting; it illustrates the theorem's form across dimensions but does not exhibit a nontrivial repositioning or construct an M-position. This is another derived calculation, not a worked example in the original preprint.

Structural Tensions¶

Immediate fixed-position calculation versus uniform repositioned guarantee. Keeping the original coordinates makes the sum immediate to define, but the perpendicular ellipses make a universal reverse bound impossible there. Allowing a determinant-one map restores a dimension-free existence guarantee at the cost of finding or proving a suitable position. Diagnostic: is the estimate about the original pair or a permitted relative image?^[1]

Pair-specific exactness versus uniform control. A special pair may permit an exact ratio or a simple inclusion, as the two derived examples do. The theorem controls all admissible pairs and dimensions with one (C); a convenient uniform bound may be coarser than a given pair's optimum, though sharp universal constants remain a separate question. Diagnostic: which quantifiers does the reported constant actually satisfy?^[1]

Structural–Framed Character¶

Evaluative weight: the theorem says what is mathematically true under its hypotheses, not what ought to be chosen. Human-practice dependence: proof and notation are mathematical practice, but the quantified convex-body relation is not institution-dependent. Institutional origin: Milman's original preprint is a research result, with a distinct later French publication; bibliographic versions should not be silently merged. Vocabulary travel: “reverse” and “normalization” occur widely, but without centrally symmetric convex bodies, determinant-one position and root-volume control they do not name this inequality. Import versus recognition: use the quantified formula and its admissible bodies to recognize instances; a verbal resemblance to other inverse inequalities is insufficient.^[1]

Its character: structural inside convex geometry and domain-specific across the encyclopedia. A broad position-before-comparison heuristic might travel, but it is not itself an approved Prime edge from this mathematical theorem.

Structural Core vs. Domain Accent¶

The nearest live genera express two different necessities: this is a Formal Theorem, and its operands are Convex Bodies. The irreducible child core is the dimension-free, repositioned Minkowski-sum upper inequality for centrally symmetric bodies. Removing the determinant-one existential choice, the root-volume scale or the upper-bound direction changes the result.^[1]

Ellipse axes and cube facets are accents of our two examples. The theorem is not made a Prime merely by noticing a general slogan about alignment; no non-mathematical instance supplies the same volume-and-Minkowski-sum structure. Any broader invariant-preserving comparison pattern would need its own independent Prime assessment.

This entry presupposes Convex body and is a kind of Formal theorem.

There are two direct strict parents. Formal theorem is a subsumption parent: Milman's inequality is a proved mathematical statement, while that genus contains many unrelated results. Convex body is a composition/presupposes parent: two such objects are required as operands, but a theorem is not a kind of convex body. These roles are independent; neither edge duplicates the other.^[1]

Formal System is inherited through Formal Theorem and Convexity through Convex Body; direct repetition would add no necessary relationship. Transformation is nearby because a suitable map may be nontrivial, but the favorable cube pair permits (T=I) and the theorem's target identity is the positioned inequality, not generic restructuring. Other named inequalities are parallel statements, not an all-instance parent.

Relationships to Other Abstractions¶

Current abstraction Milman's reverse Brunn–Minkowski inequality Domain-specific

Parents (2) — more general patterns this builds on

Milman's reverse Brunn–Minkowski inequality is a kind of Formal theorem Domain-specific

Milman's named reverse inequality is a proved mathematical statement with hypotheses and a conclusion; other formal theorems need not concern convex bodies.
Milman's reverse Brunn–Minkowski inequality presupposes Convex body Domain-specific

The inequality necessarily operates on two centrally symmetric convex bodies; convex bodies exist without this particular theorem.

Hierarchy paths (3) — routes to 3 parentless roots

Milman's reverse Brunn–Minkowski inequality → Formal theorem → Formal System → Formalization → Representation → Abstraction

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Milman's reverse Brunn–Minkowski inequality sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Ordinary Brunn–Minkowski: a lower bound without repositioning. An unpositioned reverse bound: disproved by the perpendicular ellipses. M-position as a technical construction: not established for the identity-position cube example. A polar-volume product or reverse Santaló result: a different statement. The original's normed-space corollary: an application rather than a second concrete body-pair example. A dimension-dependent estimate: weaker than one universal (C).^[1]

References¶

[1] V. D. Milman, An inverse form of the Brunn-Minkowski inequality with applications to local theory of normed spaces, Max-Planck-Institut für Mathematik preprint MPI/SFB 85/36 (1985), Abstract and Theorem 1(a) on one-based PDF pp. 2–3 for the determinant-one, centrally symmetric body-pair reverse bound and polar companion; Theorem 1(b) and Corollary 2 on one-based PDF p. 3 for the special Euclidean-ball/normed-space context. This is the 1985 English preprint, not the distinct 1986 French note. The ellipse and cube/cross-polytope calculations in this entry are explicitly derived examples, not examples printed in the preprint. https://archive.mpim-bonn.mpg.de/id/eprint/3762/1/preprint_1985_36.pdf registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x