On the smallest L_2 projection of a curve in R^n

TL;DR

This paper investigates the minimal L_2 projection of a curve in R^n by analyzing the directions in which the curve is least aligned, based on the L_2 norm of its differential.

Contribution

It introduces a method to quantify the directions a curve in R^n most avoids, using the L_2 norm of its derivative, providing new insights into curve projections.

Findings

Identifies directions minimally aligned with the curve based on L_2 differential norms

Provides a quantitative measure of how a curve 'misses' certain directions

Establishes bounds on the minimal projection in terms of the L_2 norm

Abstract

For a curve T:[0,1] -> R^n, we consider the directions theta in R^n which T "misses" the most and quantify this, as a function of the L_2 norm of T's differential.