Equality in Brunn–Minkowski for convex bodies: homothety #
Let K and L be compact convex sets of positive volume in a finite-dimensional Euclidean space
and let 0 < θ < 1. The multiplicative Brunn–Minkowski inequality
volume K ^ (1 - θ) * volume L ^ θ ≤ volume ((1 - θ) • K + θ • L) is an equality exactly when L
is a translate of K. For c, d > 0 and n = finrank ℝ E ≥ 1, the additive inequality
c * volume A ^ (1 / n) + d * volume B ^ (1 / n) ≤ volume (c • A + d • B) ^ (1 / n) is an
equality exactly when B = v +ᵥ r • A for a vector v and a scalar r > 0. Taking c = 1 - θ
and d = θ gives the weighted form, and c = d = 1 the unweighted form.
The proof completes the argument of Kneser and Süss. Under multiplicative equality, the height
quantiles of K and L along each unit vector u differ by a constant
(Convex.exists_quantile_map_inner_cond_eq_add_of_volume_le). Averaging over the levels identifies
this constant as ⟪u, v⟫, where v is the difference of the centroids of L and K
(TauCeti.setIntegral_quantile_map_inner_cond). Since the quantiles sweep out the heights of
interior points, interior L and v +ᵥ interior K have the same heights along every unit
vector, so by separation of points from open convex sets they coincide
(Convex.subset_of_image_inner_subset). Passing to closures gives
L = v +ᵥ K. The additive equality case reduces to the multiplicative one by rescaling A and B
to unit volume.
The hypothesis of positive volume is equivalent to nonempty interior
(Convex.addHaar_pos_iff_interior_nonempty).
Main results #
Convex.exists_interior_eq_vadd_interior_of_volume_le: for bounded convex sets of positive volume, multiplicative equality forces the interiors to be translates.Convex.volume_smul_add_smul_eq_iff: for compact convex sets of positive volume, multiplicative equality holds exactly for translates.Convex.volume_real_smul_add_smul_rpow_eq_iff: for compact convex sets of positive volume, additive equality holds exactly for homothetic sets.
References #
- H. Kneser and W. Süss, Die Volumina in linearen Scharen konvexer Körper, Mat. Tidsskr. B (1932), 19–25.
- R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014, Theorem 7.1.1 and its proof.
Translated interiors under Brunn–Minkowski equality. Let K and L be bounded convex sets
of positive volume and 0 < θ < 1. If the volume of (1 - θ) • K + θ • L is at most
volume K ^ (1 - θ) * volume L ^ θ, that is, if the multiplicative Brunn–Minkowski inequality is
an equality, then the interior of L is a translate of the interior of K.
Brunn–Minkowski equality for convex bodies, multiplicative form. Let K and L be compact
convex sets of positive volume and 0 < θ < 1. If the multiplicative Brunn–Minkowski inequality
volume K ^ (1 - θ) * volume L ^ θ ≤ volume ((1 - θ) • K + θ • L) is an equality, then L is a
translate of K.
Equality cases of multiplicative Brunn–Minkowski for convex bodies. Let K and L be
compact convex sets of positive volume and 0 < θ < 1. Then
volume ((1 - θ) • K + θ • L) = volume K ^ (1 - θ) * volume L ^ θ exactly when L is a translate
of K.
Brunn–Minkowski equality for convex bodies, additive form. Let A and B be compact convex
sets of positive volume in a space of dimension n, and let c, d > 0. If the reverse bound
volume (c • A + d • B) ^ (1 / n) ≤ c * volume A ^ (1 / n) + d * volume B ^ (1 / n) holds,
then B is homothetic to A: B = v +ᵥ r • A for some vector v and some r > 0.
In positive dimension, the bound gives equality in the Brunn–Minkowski inequality;
in dimension zero, homothety follows from nonempty A and B alone.
Equality cases of Brunn–Minkowski for convex bodies. Let A and B be compact convex sets
of positive volume in a space of positive dimension n, and let c, d > 0. Then
volume (c • A + d • B) ^ (1 / n) = c * volume A ^ (1 / n) + d * volume B ^ (1 / n) exactly when
B is homothetic to A, that is, B = v +ᵥ r • A for some vector v and some r > 0.