An inequality for sections and projections of a convex set

Let K ⊂ R d K \subset {{\mathbf {R}}^d} be a convex body, γ \gamma its center of mass. For Λ ⊂ R d \Lambda \subset {{\mathbf {R}}^d} a subspace of dimension d − k d - k , we establish the inequality \[ Vol d ( K ) ⩽ Vol d − k ( K | Λ ) Vol k ( ( K − γ ) ∩ Λ ⊥ ) {\operatorname {Vol} _d}(K) \leqslant {\operatorname {Vol} _{d - k}}(K|\Lambda ){\operatorname {Vol} _k}((K - \gamma ) \cap {\Lambda ^ \bot }) \] (where K | Λ K|\Lambda denotes orthogonal projection of K K onto Λ \Lambda ). Equality holds only if each k k -dimensional section of K K parallel to Λ ⊥ {\Lambda ^ \bot } is a translate of ( K − γ ) ∩ Λ ⊥ (K - \gamma ) \cap {\Lambda ^ \bot } .