An inequality for sections and projections of a convex set

Abstract:

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