  We compute the explicit sharp constants of Hardy inequalities in the cone
$\mathbb{R}_{k_+}^{n}:=\mathbb{R}^{n-k}\times
(\mathbb{R}_{+})^{k}=\{(x_{1},...,x_{n})|x_{n-k+1}>0,...,x_{n}>0\}$ with $1\leq
k\leq n$. Furthermore, the spherical harmonic decomposition is given for a
function $u\in C^{\infty}_{0}(\mathbb{R}_{k_+}^{n})$. Using this decomposition
and following the idea of Tertikas and Zographopoulos, we obtain the
Filippas-Tertikas improvement of the Hardy inequality.
