## Abstract

We consider planar central configurations of the Newtonian 2*n*-body problem consisting in two twisted regular *n*-gons of equal masses. We prove the conjecture that for \(n\ge 5\) all convex central configurations of two twisted regular *n*-gons are strictly convex.

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## References

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## Additional information

This work has been realized thanks to the MINECO Grants MTM2016-80117-P and MTM2016-77278-P (FEDER) and Catalan (AGAUR) Grants 2017 SGR 1374 and SGR 1617.

## Appendix

### Appendix

### Proof of Lemma 1

We want to prove that, for any fixed value \(a\in [\cos (\pi /7),1)\), the function

is positive in the closed interval \(u\in {{\mathscr {K}}}=\left[ u_m(a),\dfrac{3a}{b}\right] \), where \(b=\sqrt{1-a^2}\) and

The function *T* is clearly continuous and differentiable in the domain, so *T* must attain an absolute minimum on \({\mathscr {K}}\). On one hand, if we look for critical points in the interior of \({\mathscr {K}}\),

occurs at \(u_c=(3a+\sqrt{a^2+8})/(4b)\). It is not difficult to see that \(u_c<u_m\) for \(a\in [\cos (\pi /7),1)\), and \(\partial T/\partial u (a,u) >0\) for \(u\in [u_m,3a/b]\). Therefore, the minima of *T* occurs at \(u=u_m(a)\).

To conclude the proof it is enough to see that \(T(a,u_m(a))>0\). Simplifying, we want to see that

which is equivalent to prove that

Squaring both sides, the expression factors into

where *p*(*x*) is a polynomial of degree 13 with integer coefficients, which using Sturm’s Theorem contains no real zeros in the interval [9 / 10, 1].

### Proof of Lemma 2

We want to prove that, for any fixed value of \(a\in [\cos (\pi /6),1)\), the function

is negative for any \(u\in {{\mathscr {K}}}=\left[ \dfrac{a}{b},u_m(a)\right] \), where \(b=\sqrt{1-a^2}\) and

The function is clearly continuous and differentiable in \({\mathscr {K}}\). On one hand, the function *T*(*a*, *u*) is strictly increasing with respect the variable *u* because

Therefore, \(T(a,u) \le T(a,u_m(a))\) and

Finally, it is not difficult to see that \(p'(a)\) has no roots in \(a\in [\cos (\pi /6),1]\) (for example, using Sturm’s theory) and \(p'(a)>0\). Therefore, \(T(a,u_m(a)) < p(1) =0\), which concludes the proof.

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Barrabés, E., Cors, J.M. On Strictly Convex Central Configurations of the 2*n*-Body Problem.
*J Dyn Diff Equat* **31, **2293–2304 (2019). https://doi.org/10.1007/s10884-018-9708-5

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### Keywords

- Central configuration
- Convex central configuration
*n*-Body problem- Twisted central configuration

### Mathematics Subject Classification

- 70F10
- 70F15