The classification of 3- and 4- Helly-dimensional convex bodies

A new step in the solution of the Szökefalvi-Nagy problem

“…One of them (Lemma 15) is evident, and the others are known in the theory of convex sets. A similar construction is contained in Kincses' paper [9].…”

“…In [8] Boltyanski and Chabukiani have listed all convex bodies in R 3 that satisfy the condition him T(M) = 2 (without the assumption of central symmetry). In [9] Kincses has listed all centrally symmetric convex bodies in R d that satisfy the conditions him T(M) = 3 or him T(M) = 4. Furthermore, we mention one more result [2], [10] that gives a formal algebraic solution of the Sz6kefavi-Nagy problem in the general case.…”

“…However, we use a new approach to a solution of the Szrkefalvi-Nagy problem. This approach (applied in [3], 14], [81 and, in particular, [9]) involves describing not the bodies M with md M = r, but their polar bodies M* instead.…”

Solution of the Sz�kefalvi-Nagy problem for a class of convex polytopes

“…Partial geometrical results in the direction of a solution of the Sz6kefalvi-Nagy problem were established in [4], [5], [6], [8]. We indicate the following theorem of Boltyanski.…”

Solution of the Illumination Problem for Bodies with md M = 2

“…The articles [1]- [3], [11], and [25] contain further results on the Szökefalvi-Nagy problem. Some other applications of md M are given in Section 44 of [17].…”