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

TL;DR

This paper characterizes the lines that minimize the L^p-distance to the vertices of an equilateral triangle across all p values, revealing a transition in optimal line configurations.

Contribution

It provides a complete geometric description of L^p-optimal lines for all p in [1,∞], including special cases and families of solutions.

Findings

Optimal lines are parallel to triangle sides for 1<= p < 4/3 and 2<p<=∞.

Perpendicular bisectors are optimal for 4/3<p<2.

For p=2 and p=4/3, there are continuous families of optimal lines.

Abstract

We consider the distances between a line and a set of points in the plane defined by the L^p-norms of the vector consisting of the euclidian distance between the single points and the line. We determine lines with minimal geometric 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,\infty establishes the passage between the well-known sets of optimal lines for p=1,2,\infty. The set of optimal lines consists of three lines each parallel to one of the triangle sides for 1<= p < 4/3 and 2<p<=\infty and of the three perpendicular bisectors of the sides for 4/3<p<2. For p=2 and p=4/3 there exist one-dimensional families of optimal lines.