Enclosing a convex body by homothetic bodies (Q791158)

Let \(B\) be a convex body in \(E^ d\) and let \(S(B)\) denote its surface area. Consider homothetic copies of \(B\), say \(B_ 1,\dots,B_ n\), having no interior points in common with \(B\), and such that any continuous curve connecting a point of \(B\) with a point outside of \(\text{conv}(B_ 1\cdots B_ n)\) intersects at least one \(B_ i\). For such enclosing families the authors investigate \(\sum S(B_ i)/S(B)\) and relate it to an affine invariant \(M(B)\) of \(B\). The functional \(M(B)\) is defined as \(\int(1/F)\,dS\), where the integral is extended over the surface of \(B\), and \(F\) is the \((d-1)\)-dimensional volume of the section of \(B\) maximal \((d-a)\)-dimensional volume with a hyperplane parallel to the supporting hyperplane of \(B\) at the surface area element \(dS\). Theorem 1. For \(d=2\) we have \(\sum S(B_ i)/S(B)\geq M(B)\). Equality holds if and only if \(B\) is a parallelogram. Theorem 2. For \(d\geq 2 \inf \sum S(B_ i)/S(B)\leq M(B)\), where the infimum is extended over all enclosing families of \(B\). The authors also investigate the interesting related problem of covering the surface of \(B\) by finite families of homothets of \(B\).