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

Carathéodory's Theorem and H-Convexity

“…if n 3 and md H=2, then n&1 ctd (H ) n 2 &n&1, (4) if n 4 and 3 md H n&1, then n&1 ctd (H ) , (5) if md H=n 3, then n ctd (H) .…”

“…Since H is not splittable, the body M is indecomposdable, i.e., it is not representable as the direct vector sum of two convex sets with positive dimensions. In [4] (cf. also Theorem 29.1 in [9]) it is proved that there are only four types of compact, convex, indecomposable bodies M/R n with md H(M )=2, n 3: stacks, outcuts, stack-outcuts, and particular four-dimensional polytopes.…”

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

“…This problem is solved in [15] and [12] for m = 2. The following theorem [15] describes all three-dimensional bodies with md M = 2; the stacks and the outcuts mentioned in its statement are defined below.…”

“…In this section we formulate the solution of the Szökefalvi-Nagy problem for the case md M = 2 given in [12]. We explain this solution here, since it is used below in the proof of the Main Theorem.…”

“…A polytope M ⊂ R 4 containing the origin in its interior (and any its translate) is said to be a particular four-dimensional polytope if there is a basis e 1 , e 2 , e 3 , e 4 of R 4 The proof is given in [12].…”