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We consider the distances between a line and a set of points in the planedefined by the L^p-norms of the vector consisting of the euclidian distancebetween the single points and the line. We determine lines with minimalgeometric L^p-distance to the vertices of an equilateral triangle for all 1<=p<=\infty. The investigation of the L^p-distances for p\ne 1,2,\inftyestablishes the passage between the well-known sets of optimal lines forp=1,2,\infty. The set of optimal lines consists of three lines each parallel toone of the triangle sides for 1<= p < 4/3 and 2<p<=\infty and of the threeperpendicular bisectors of the sides for 4/3<p<2. For p=2 and p=4/3 there existone-dimensional families of optimal lines.

Geometrically L^p-optimal lines of vertices of an equilateral triangle