More on Compacta with Convex Projections

Abstract

We obtain information about the structure of nonconvex compacta $C$ in $\R^n$ ($n\ge3$) having the property that every projection onto a hyperplane is convex. We find that if $\dim(C)\le n-2$, then $C$ contains $n+1$ copies of $S^{n-2}$ that are contained in distinct hyperplanes. If $\dim(C)\ge n-1$, then the existence of three and no more than three hyperplanes that intersect $C$ in an $(n-2)$-sphere can be guaranteed.