The classification of 3- and 4-Helly-dimensional convex bodies
The Helly dimension of a convex body K is the smallest integer k such that whenever \({\mathcal S}\) is a finite collection of translates of K with the property that every \(k+1\) bodies in \({\mathcal S}\) have a non-empty intersection then the intersection of all bodies in \({\mathcal S}\) is non- empty. \textit{L. Nachbin} [Trans. Am. Math. Soc. 68, 28-46 (1950; Zbl 0035.354)] and \textit{J. L. Kelley} [ibid. 72, 323-326 (1952; Zbl 0046.120)] proved: A centrally-symmetric convex body has Helly dimension 1 if and only if it is the direct sum of line segments (1-dimensional convex bodies) i.e., a parallelotope. \textit{B. Sz.-Nagy} [Acta Sci. Math. 15, 169-177 (1954; Zbl 0057.145)] proved this with ''centrally-symmetric'' deleted. \textit{V. G. Boltyanskij} [Sov. Math., Dokl. 17, 674-677 (1976), translation from Dokl. Akad. Nauk. SSSR 228, No.2, 265-268 (1976; Zbl 0359.52008)] proved: A centrally-symmetric convex body has Helly dimension 2 if and only if it is the direct sum of one- and two-dimensional convex bodies. Counterexamples show that such a simple direct sum description is not possible for Helly dimension greather than 2. In the present paper the author gives a complete classification of centrally-symmetric bodies with Helly dimension 3 and 4. The proof is based on a theorem proved by \textit{I. Bárány} and the author [''Characterization of k-Helly dimensional convex bodies'', to appear].
classification of centrally-symmetric bodies with Helly dimension 3 and 4
scientific article; zbMATH DE number 3992329