Approximation under an arc length constraint

Abstract

Let f ∈ C[a, b]. LetP be a subset ofC′[a, b], L ≥b − a be a given real number. We say thatp ∈ P is a best approximation tof fromP, with arc length constraintL, ifA[p] ≡ ∫ba√[1 + (p′(x))2]dx ≤ L and∥p − f∥ ≤ ∥q − f∥ for allq ∈ P withA[q] ≤ L. ∥⋅∥ represents an arbitrary norm onC[a, b]. The constraintA[p] ≤ L might be interpreted physically as a materials constraint.