]> A Practical Guide to Tensegrity Design: B Proof that the Constraint Region is Non-convex

Appendix B

Proof that the Constraint Region is Non-convex

In Section 3.2, the claim is made that the constraint region in the general tensegrity mathematical programming problem is not convex. A proof is given here.

The non-convexity is due to the strut constraints. To demonstrate this, let Pa and Pb be one set of admissible end points for a strut which meet its length constraint with equality, and let P a' and P b' be another such set. Let l n be the value of the corresponding constraint constant. By assumption:

Pa -Pb2 = l n2
P a' -Pb' 2 = l n2

Let P a'' and P b'' be a convex combination of these two point sets. This means:

P a'' λ Pa+ 1-λ Pa'
P b'' λ Pb+ 1-λ Pb'

where λ 01. Therefore:

P a'' -Pb'' 2 = λ Pa - Pb + 1-λ Pa' - Pb' 2
P a'' -Pb'' 2 = λ2 Pa -Pb2 +2λ 1-λ Pa -Pb Pa' - Pb' + 1-λ 2 P a' -Pb' 2

By the Schwarz inequality:1

Pa -Pb Pa' - Pb' < Pa -Pb P a' -Pb'

So:

P a'' -Pb'' 2 < λ2 Pa -Pb2 +2λ 1-λ Pa -Pb Pa' - Pb' + 1-λ 2 P a' -Pb' 2
P a'' -Pb'' 2 = λ Pa - Pb + 1-λ Pa' - Pb' 2
P a'' -Pb'' 2 = λ l n + 1-λ l n 2
P a'' -Pb'' 2 = l n2

In summary:

l n2 > P a'' -Pb'' 2

This means the constraint is violated; hence, the constraint region is not convex.


1 See Lang71, p. 22. This is also known as the Cauchy-Schwarz inequality or the Cauchy-Buniakovskii-Schwarz inequality.