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

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

A Practical Guide to Tensegrity Design
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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 $P_{a}$ and $P_{b}$ 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 ${\overline{l}}_{n}$ be the value of the corresponding constraint constant. By assumption:

${|P_{a} - P_{b}|}^{2} = {\overline{l}}_{n}^{2}$
${|P_{a'} - P_{b'}|}^{2} = {\overline{l}}_{n}^{2}$

Let $P_{a''}$ and $P_{b''}$ be a convex combination of these two point sets. This means:

$P_{a''} \equiv λ P_{a} + (1 - λ) P_{a'}$
$P_{b''} \equiv λ P_{b} + (1 - λ) P_{b'}$

where $λ \in (0, 1)$ . Therefore:

{|P_{a''} - P_{b''}|}^{2} = {|λ (P_{a} - P_{b}) + (1 - λ) (P_{a'} - P_{b'})|}^{2}

= λ^{2} {|P_{a} - P_{b}|}^{2} + 2 λ (1 - λ) (P_{a} - P_{b}) \cdot (P_{a'} - P_{b'}) + {(1 - λ)}^{2} {|P_{a'} - P_{b'}|}^{2}

By the Schwarz inequality:¹

$(P_{a} - P_{b}) \cdot (P_{a'} - P_{b'}) < |P_{a} - P_{b}| |P_{a'} - P_{b'}|$

So:

{|P_{a''} - P_{b''}|}^{2} < λ^{2} {|P_{a} - P_{b}|}^{2} + 2 λ (1 - λ) |P_{a} - P_{b}| |P_{a'} - P_{b'}| + {(1 - λ)}^{2} {|P_{a'} - P_{b'}|}^{2}

= {(λ |P_{a} - P_{b}| + (1 - λ) |P_{a'} - P_{b'}|)}^{2}

= {(λ {\overline{l}}_{n} + (1 - λ) {\overline{l}}_{n})}^{2}

= {\overline{l}}_{n}^{2}

In summary:

${\overline{l}}_{n}^{2} > {|P_{a''} - P_{b''}|}^{2}$

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.

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