The Steiner Ratio of L _p -planes

Abstract

We consider the Steiner Problem in L ² _p , which is a plane equiped with the p-norm. Steiner’s Problem is the “Problem of shortest connectivity”, that means, given a finite set N of points in the plane, search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST) for N.

Steiner’s Problem is very hard as well in combinatorial as in computational sense, but on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacI % cacaaIYaGaaiilaiaadchacaGGPaGaaiOoaiabg2da9iGacMgacaGG % UbGaaiOzaiaacUhadaWcaaqaaiaadYeacaGGOaGaam4uaiaad2eaca % WGubGaaGjcVlaadAgacaWGVbGaamOCaiaayIW7caWGobGaaiykaiaa % ygW7caaMb8UaaGzaVlaaygW7caaMb8UaaGzaVdqaaiaadYeacaGGOa % Gaam4uaiaad2eacaWGubGaaGjcVlaadAgacaWGVbGaamOCaiaayIW7 % caWGobGaaiykaaaacaGG6aGaamOtaiaayIW7cqGHgksZcaWGmbWaa0 % baaSqaaiaadchacaaMi8oabaGaaGOmaaaakiaayIW7caWGPbGaam4C % aiaayIW7caaMi8UaamyyaiaayIW7caWGMbGaamyAaiaad6gacaWGPb % GaamiDaiaadwgacaaMi8Uaam4CaiaadwgacaWG0bGaaiyFaaaa!7DF1!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$m(2,p): = \inf \{ \frac{{L(SMT{\kern 1pt} for{\kern 1pt} N)}}{{L(SMT{\kern 1pt} for{\kern 1pt} N)}}:N{\kern 1pt} \subseteq L_{p{\kern 1pt} }^2{\kern 1pt} is{\kern 1pt} {\kern 1pt} a{\kern 1pt} finite{\kern 1pt} set\} $$

which is called the Steiner ratio (of L ² _p ).

We look for estimates for m(2, p), depending on the parameter p, and, on the other hand, we will determine general upper bounds for the Steiner ratio of L ² _p .