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 bounds the volume of a convex-body sum after the bodies are suitably positioned. For centrally symmetric, full-dimensional compact convex bodies (K,L\subset\mathbb R^n), Milman's original theorem supplies a numerical (C), independent of the dimension and body shapes, and a linear map (T) with determinant (1) such that

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

Here (K+T(L)) contains all sums of one point from each body. The (n)-th root puts each volume on a length scale, and determinant (1) leaves (|L|) unchanged. The bound holds after some suitable relative map. It is not claimed for every original orientation.[^ref-db931ea93890]

Scope of Application¶

This is a theorem in convex geometry for the symmetric convex-body class in the cited original preprint. Normed-space unit balls are one reason such a theorem matters, but the preprint's polar-body companion and normed-space corollary are separate statements. The ellipse and cube examples below are elementary calculations derived for this entry and checked independently; they were not worked in Milman's preprint.[^ref-db931ea93890]

Clarity¶

Ordinary Brunn–Minkowski lower-bounds a sum's volume radius without repositioning. Milman's reverse upper-bounds it, up to a universal constant, after a suitable volume-preserving relative repositioning. State the inequality direction and whether the map has already been chosen. A special pair's exact constant is not the theorem's optimal universal constant.[^ref-db931ea93890]

The displayed form has unit coefficients. This entry does not infer a formula for arbitrary weights, a technical M-position for the examples, or a reverse Santaló result from their calculations.[^ref-db931ea93890]

Manages Complexity¶

Equal-volume bodies can be elongated in incompatible directions, making their sum much larger than either input. The theorem separates body size, measured by volume radius, from relative position, which may be changed by a determinant-one map. The difficult guarantee is that a favorable position exists with one constant across dimensions and shapes.[^ref-db931ea93890]

Abstract Reasoning¶

First check that the bodies meet the symmetry, convexity and positive-volume hypotheses. Compare (|K|^{1/n}+|L|) with the volume radius of their sum. If a universal upper claim is made in the initial position, test long thin bodies at right angles. Then ask whether a determinant-one map can realign one body while preserving its volume. In an easy nested pair, identity may already give a bound; that does not remove the theorem's position choice for arbitrary pairs.[^ref-db931ea93890]

Knowledge Transfer¶

The two worked pairs use the same test with different geometry. The ellipse pair needs a nontrivial correction, while the cube/cross-polytope pair is already favorably placed in every dimension (n\ge2). Their calculations help explain the theorem's roles but do not prove it for all symmetric convex bodies.[^ref-db931ea93890]

The live Formal Theorem entry is a strict genus because this is a proved mathematical statement. The live Convex Body entry is a strict prerequisite because its objects are the operands. One relation concerns the kind of statement; the other concerns its necessary objects. The theorem remains a specific mathematical identity, not a generic “align before comparing” rule.

Example¶

Let (a>1), and take the ellipses (K={(x,y):x^2/a2+a^2y2\le1}) and (L={(x,y):a^2x2+y^2/a2\le1}). Each has area (\pi), but their long axes are perpendicular. Before repositioning, (K+L) contains the square ([-a,a]^2), so its area radius is at least (2a), while the input radius sum stays (2\sqrt\pi). The ratio grows with (a). The determinant-one map (T=\operatorname{diag}(a^2,a)) sends (L) to (K); then (K+T(L)=2K) and the ratio is exactly (1). Mapped back: symmetric bodies, bad initial position, volume-preserving correction, bounded sum. This is our derived example, not a printed example in Milman's preprint.[^ref-db931ea93890]

For an unlike polyhedral case in every (n\ge2), take (K=[-1,1]^n) and (L={x:\sum_i|x_i|\le1}), a cross-polytope. Since (L\subset K), identity gives (K+L\subset2K), hence (|K+L|^{1/n}\le4). The input radius sum is (2+2/(n!)^{1/n}\ge2), so the ratio is at most (2) for all such (n). Mapped back: the same theorem roles are filled with (T=I); nesting makes this pair easy. It does not establish identity as a technical M-position or prove the universal theorem. This too is a derived calculation.[^ref-db931ea93890]

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¶

An unpositioned reverse bound fails for the perpendicular ellipses. Ordinary Brunn–Minkowski is a lower bound. The cube inclusion is an easy special case, not a universal proof. M-position, polar-volume results and reverse Santaló require their own precise statements. A bound with a factor depending on dimension is weaker than Milman's dimension-free guarantee.[^ref-db931ea93890]

References¶

[^ref-db931ea93890]: 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