Shadows of convex bodies

Abstract

It is proved that if $C$ is a convex body in ${\mathbb{R}}^n$ then $C$ has an affine image $\tilde{C}$ (of non-zero volume) so that if $P$ is any 1-codimensional orthogonal projection, $$|P\tilde{C}|\ge |\tilde{C}{|}^{\frac{n-1}{n}}.$$ It is also shown that there is a pathological body, $K$, all of whose orthogonal projections have volume about $\sqrt{n}$ times as large as $|K{|}^{\frac{n-1}{n}}$.