Gruber P. Convex and Discrete Geometry

8 The Brunn–Minkowski Inequality 163
h � (1 − λ)x + λy + � (1 − λ)s + λt � u � ≥ f (x + su) 1−λ g(y + tu) λ for x, y ∈ E d−1 .
The induction hypothesis thus implies that for the functions F, G, H defined by
�
�
�
F(s) = f (x + su) dx, G(t) = g(y + tu) dy, H(r) = h(z + ru) dz
E d−1
E d−1
for r, s, t ∈ R, the following statement holds:
E d−1
F, G, H are non-negative measurable functions on R such that
H � (1 − λ)s + λt � ≥ F(s) 1−λ G(t) λ for s, t ∈ R.
The case d = 1 of the Prékopa–Leindler inequality, applied to F, G, H and Fubini’s
theorem then yield inequality (1) for d. The induction and thus the proof of the
theorem is complete. ⊓⊔
A Multiplicative Version of the Brunn–Minkowski Inequality
As a consequence of the Prékopa–Leindler inequality we have the following result.
Theorem 8.15. Let S, T be non-empty compact sets in E d . Then
V � (1 − λ)S + λT � ≥ V (S) 1−λ V (T ) λ for 0 ≤ λ ≤ 1.
Proof. We may assume that 0