An Introduction to Transformational Geometry by F.M. Eccles

By F.M. Eccles

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I . / IB in the corresponding d-dimensional linear space AClaccording t o the so-called Miriltomski fui~ctiorlal: 1 1 ~ 1 =1 i~ n f { t > 0 :v E t B ) for any3uin Ad \{o), and On the other hand. 11 be a norm in Ad. 1 : ~4~ + is a real-valued function satisfying (i) positivity: 1 . c / > 0 for any 'L' in ACL; his is the specific form of Steiner's Problem for two given points (ii) identity: I vli = 0 if and only if 2: = o; (iii) and homogeneity: (ltvll = ltl . i/vil for any v in Ad and any real t: (iv) triangle inequality: Iv + vll 5 Ilv / + i l ~ i ' l l for any w ,v' in Ad.

The first survey of Steiner's Problem in the Euclidean plane was presented by Gilbert and Pollak in 1968 [186]; they christened the terms "Steiner Minimal Tree" for the shortest inkrconnecting network and "Steiner points" for the additional vertices. I t is u-ell-known that solutions of network design problems depend essentially on the wag in which the distances in space are determined. Clearly, this is true for Steiner's Problem. Consequently, there are many metric spacesg to be considered. Surveys in form of monographs are given by 1.

Proof. The equation ( 1 . 1 1 ) also holds true in d dimensions. Hence we have that is, an inequality which is satisfied only for 77, 5 3. For the planar case we know more about the ~ P r t e xdegrees. 21 Conszder S M T s i n a Banach planes equipped with u unit ball B . T h e n ( a ) ( C . [gl], Swanepoel [4lG]) For the degrees of the vertices the following holds true: If B is a n ajjinely regular hezagon, then the degree is at most 6, otherwise at m o s t 4 . al. 5 STEINER'S PROBLEM I N GRAPHS Connectivity is also a very important concept in combinatorial optimization.

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