To complement Suvrit's comment, the set of $n \times n$ symmetric Toeplitz matrices is parameterized by
$$\left\{ x_1 \mathrm M_1 + x_2 \mathrm M_2 + \cdots + x_n \mathrm M_n \mid x_1, x_2, \dots, x_n \in \mathbb R \right\}$$
where $\mathrm M_1, \mathrm M_2, \dots, \mathrm M_n$ are $n \times n$ symmetric Toeplitz basis matrices. Let $\mathrm M_1 = \mathrm I_n$ correspond to the main diagonal, whereas the remaining basis matrices correspond to super and supra diagonals.
Let $\mathrm M : \mathbb R^n \to \mbox{Sym}_n (\mathbb R)$ be defined by
$$\mathrm M (\mathrm x) := x_1 \mathrm M_1 + x_2 \mathrm M_2 + \cdots + x_n \mathrm M_n$$
Using the spectral norm, we obtain the following optimization problem in $\mathrm x \in \mathbb R^n$
$$\begin{array}{ll} \text{minimize} & \| \mathrm M (\mathrm x) - \mathrm A \|_2\end{array}$$
which can be rewritten as the following semidefinite program (SDP) in $\mathrm x \in \mathbb R^n$ and $t \geq 0$
$$\boxed{\begin{array}{ll} \text{minimize} & t\\ \text{subject to} & - t \,\mathrm I_n \preceq - \mathrm A + x_1 \mathrm M_1 + x_2 \mathrm M_2 + \cdots + x_n \mathrm M_n \preceq t \,\mathrm I_n\end{array}}$$