  Let $\omega\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary.
For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone
$\Omega_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n:
x'\in\varepsilon x_1\omega\big\}\subset\mathbb{R}^{n+1}$ and the operator
$Q_{\varepsilon}^{\alpha}$ acting as the Laplacian $u\mapsto-\Delta u$ on
$\Omega_{\varepsilon}$ with the Robin boundary condition $\partial_\nu u=\alpha
u$ at $\partial\Omega_\varepsilon$, where $\partial_\nu$ is the outward normal
derivative and $\alpha>0$. We look at the dependence of the eigenvalues of
$Q_\varepsilon^\alpha$ on the parameter $\varepsilon$: this problem was
previously addressed for $n=1$ only (in that case, the only admissible $\omega$
are finite intervals). In the present work we consider arbitrary dimensions
$n\ge2$ and arbitrarily shaped "cross-sections" $\omega$ and look at the
spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes
"sharp" and collapses to a half-line. It turns out that the main term of the
asymptotics of individual eigenvalues is determined by the single geometric
quantity $N_\omega:=\dfrac{\mathrm{Vol}_{n-1} \partial\omega }{\mathrm{Vol}_n
\omega}$. More precisely, for any fixed $j\in \mathbb{N}$ and $\alpha>0$ the
$j$th eigenvalue $E_j(Q^\alpha_\varepsilon)$ of $Q^\alpha_\varepsilon$ exists
for all sufficiently small $\varepsilon>0$ and satisfies
$E_j(Q^\alpha_\varepsilon)=-\dfrac{N_\omega^2\,\alpha^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$
as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces
on infinite cones, which can be of independent interest.
