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244 Boglárka Tóth and L.G. Casado2.2 Simplification using the Baumann pointIn this section we will see that the use of the Baumann point [1], in place of the center c,simplify our computations. For a given box X = (X 1 , X 2 ), if the enclosure of the derivative off, i.e. G(X) = ([g1 L, gU 1 ], [gL 2 , gU 2 ]) is given, the Baumann point b is defined as:⎧gi ⎪⎨UxL i − gL i xU i, if ggb i = iU − giL i L < 0 < giU i = 1, 2 (6)x ⎪⎩U i if gi U ≤ 0x L i if gi L ≥ 0In the following theorem it is shown that using the Baumann point the different intersectioncases between OB and PR can be avoided from the case analysis.Theorem 1. Suppose a given differentiable function f : R 2 → R, its inclusion function F and theinclusion function of its derivative G(X) = ([g1 L, gU 1 ], [gL 2 , gU 2 ]) over a given box X. The pruneableregion defined by its vertices as in (1-2) centered in the Baumann point (i.e. c = b) include all thecorners of X, or none of them.Corollary 2. From the fact, that xU i −b i= xL pr U i −b i, ∀i, it can be deduced easily that the shifted pruneablei pr L ibox SP B ob U is exactly the same as SP Biob L, thus it can be denoted as SP Bii . Therefore, in the shiftedcases there are only two new generated boxes, and only the direction to shift have to be determined.( ) ( )Remark 3. The above results suggest to denote for i = 1, 2 the xU i −b i= xL pr U i −b i, i.e. obUi pr L i= obLi pr U ii pr L ivalues and the 1− xU i −b i, i.e. 1− obU pr U ivalues by sfipr U ii(Shifting Factor) and osf i(Opposite Shifting Factor),respectively. Therefore, A(SP B i ) = sf iosf iA(BP R).Theorem 4. If CP B ⊄ OB, then exists SP B such that A(SP B) > A(CP B ∩ OB).The advantages of the usage of the Baumann point are that it makes our computation easier,and it equilibrates the pruneable region above the box.3. The n-dimensional caseNow we generalize the above results for multi-dimensional case. As in the two-dimensionalcase let us center the problem at the point c. Thus, we will use the same notation for theOriginal Box OB = X − c.It is easy to see that, similarly to the two-dimensional case, but already re-centered in c, thevertices are:vi L = (0, . . . , 0, pr L i , 0, . . . , 0), vU i = (0, . . . , 0, pr U i , 0, . . . , 0), i = 1, . . . , n, (7)= ˜f−F L (c)g L i, pr L i = ˜f−F L (c), i = 1, . . . , n.giUwhere pr U iFrom (7) we know that in a 3-dimensional case the shape of PR is as in Figure 6. To see theproperties of this body, we do a small geometrical evasive.Definition 5. An n-dimensional polytope is called orthopeder, if the diagonals intersect in one pointand are orthogonal to each other.Proposition 6. The PR defined by its vertices (7) is an orthopeder.Definition 7. The cross polytope is the regular polytope in n dimensions corresponding to the convexhull of the points formed by permuting the coordinates (±1, 0, 0, . . . , 0).