]> 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:

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

By the Schwarz inequality:1

$Pa -Pb ⋅ Pa' - Pb' < Pa -Pb ⁢ P a' -Pb'$

So:

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

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.