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Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying $$ \left| f_{\mathbf{x}}(t) \right|\le 1, \forall t\in[0,1], $$ where the function $f_{\mathbf{x}}(t)\triangleq a_0+\sum_{m=1}^n \left(a_m\cos(2\pi mt)+b_m\sin(2\pi m t)\right)$. It is straightforward to see that $\mathcal{X}$ is convex and centrally symmetric.

My question is how to find the largest inscribed parallelepiped of $\mathcal{X}$. Comments on other geometric properties of $\mathcal{X}$ are welcome as well.

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  • $\begingroup$ This set is invariant under a one-parametric family of isometries which can be seen by substituting $t\mapsto t - t_0$. $\endgroup$
    – Maxim Prasolov
    Commented Sep 25 at 10:13

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