Maximal volume enclosed by plates and proof of the chessboard conjecture
Let C be a convex body in n-dimensional Euclidean space \(R^ n\) (n\(\geq 2)\) and let \(\Lambda\) be a lattice generated by n linearly independent vectors. The system \(\Lambda_ C\) consisting of the translates of C by the vectors of the lattice \(\Lambda\) is called the body lattice of C, with lattice \(\Lambda\). As usual, the density of \(\Lambda_ C\) is defined by \(d(\Lambda_ C)=Vol C/Vol E,\) where E is an elementary cell of \(\Lambda\). Theorem 1: Suppose that all connected components of the complementary set of \(C+\Lambda = \{c+\lambda | c\in C,\quad \lambda \in \Lambda \}\) are bounded. Then \(d(\Lambda_ C)\geq 1/2.\) If \(\Lambda\) consists of all integer points \((k_ 1,...,k_ n)\) with even sum of coordinates, and C is the unit cube \(\{(x_ 1,...,x_ n)| 0\leq x_ i\leq 1\},\) then \(\Lambda_ C\) is a chessboardlike configuration satisfying the requirements of Theorem 1 and having density 1/2. It is proved that, for \(n\geq 3\), \(d(\Lambda_ C)=1/2\) if and only if C is a parallelotope, and \(\Lambda_ C\) is affinely equivalent to the chessboard lattice (Theorem 6). Theorem 1 solves a variant of a problem proposed by \textit{L. Fejes Tóth} [Research problem 14, Period. Math. Hung. 6, 277-278 (1975)]. The proof of Theorem 1 is based on Theorem 3: Let us be given a compact set \(P=\overline{int P}\subset R^ n\), which is either the union of finitely many (non-degenerate) closed n-simplices, or is bounded by a \(C^ 1\) submanifold. Define a Borel measure \(\mu =\mu_ p\) on \(S^{n-1}\) by \(\mu (A)=\lambda_{n-1}(A')\), where A' is the set of those boundary points of P at which there exists a support hyperplane to P with outward normal vector in A; \(A\subset S^{n-1}\) is a Borel set, and \(\lambda_{n-1}\) denotes the Lebesgue measure. Let C be the unique convex body with surface area function \(\mu_ C=\mu\). Then \(Vol P\leq Vol C\) with equality if and only if \(P=C\). Let \(A_ 1,...,A_ m\) be (n-1)-dimensional polytopes (plates) in \(R^ n\). Let U be the union of all bounded components of \(R^ n\setminus \cup^{m}_{i=1}A_ i'\), where \(A_ i'\) is a translate of \(A_ i\). Theorem 3 is used to prove an inequality \(Vol U\leq Vol P,\) where P is a certain centrally symmetric convex polytope which is associated to \(A_ 1,...,A_ m\) in a natural way. In a final section some related results and conjectures are discussed. The paper makes an important contribution to the study of lattice arrangements of convex bodies.
scientific article; zbMATH DE number 3988132
scientific article