For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (ℓ + 2)-body problem with ℓ ⩾ 3. In particular, we prove that the symmetric (2n + 1)-body problem with masses m₁ = … = m_2n−1 = 1 and m_2n = m_2n+1 = m sufficiently small has at least two classes of convex central configuration when n = 2, five when n = 3, and four when n = 4. We conjecture that the (2n + 1)-body problem has at least n classes of convex central configurations for n > 4 and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric (2n + 2)-body problem with masses m₁ = … = m_2n = 1 and m_2n+1 = m_2n+2 = m sufficiently small has at least three classes of convex central configuration when n = 3, two when n = 4, and three when n = 5. We also conjecture that the (2n + 2)-body problem has at least [(n +1)/2] classes of convex central configurations for n > 5 and we give some numerical evidences that the conjecture can be true.

中文翻译：

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关于对称（ℓ+ 2）体问题的凸中心构型

对于4体问题，有以下猜想：给定任意正质量，平面4体问题对于其凸包上的质量的每个顺序都具有唯一的凸中心配置。直到现在，这个猜想一直没有解决。我们的目的是为了证明这个猜想不能扩展到（ℓ + 2） -体与问题ℓ ⩾3.特别是，我们证明了对称（2 Ñ + 1） -体质量问题米₁ = ... = m _{2 n -1} = 1和m _{2 n} = m _{2 n +1} = m足够小的具有至少两类凸中央配置的时Ñ = 2，五当Ñ = 3，以及四个当Ñ = 4我们猜想，（2 Ñ + 1） -体的问题具有至少Ñ凸出的中心的类当n > 4时，我们给出了一些数值证据，证明该猜想是正确的。我们还证明了质量为m ₁ =…= m _{2 n} = 1且m _{2 n +1} = m _{2 n +2} = m的对称（2 n + 2）体问题足够小，当n = 3时至少有三类凸中心构型，当n = 4时有两类，当n = 5时三类。我们还推测（2 n + 2）体问题至少具有[[ n + 1）/ 2]对于n > 5的凸中心构型类别，我们提供了一些数值证据，证明该猜想是正确的。