  In this paper, we consider the AWGN channel with a power constraint called
the $(\sigma, \rho)$-power constraint, which is motivated by energy harvesting
communication systems. Given a codeword, the constraint imposes a limit of
$\sigma + k \rho$ on the total power of any $k\geq 1$ consecutive transmitted
symbols. Such a channel has infinite memory and evaluating its exact capacity
is a difficult task. Consequently, we establish an $n$-letter capacity
expression and seek bounds for the same. We obtain a lower bound on capacity by
considering the volume of ${\cal S}_n(\sigma, \rho) \subseteq \mathbb{R}^n$,
which is the set of all length $n$ sequences satisfying the $(\sigma,
\rho)$-power constraints. For a noise power of $\nu$, we obtain an upper bound
on capacity by considering the volume of ${\cal S}_n(\sigma, \rho) \oplus
B_n(\sqrt{n\nu})$, which is the Minkowski sum of ${\cal S}_n(\sigma, \rho)$ and
the $n$-dimensional Euclidean ball of radius $\sqrt{n\nu}$. We analyze this
bound using a result from convex geometry known as Steiner's formula, which
gives the volume of this Minkowski sum in terms of the intrinsic volumes of
${\cal S}_n(\sigma, \rho)$. We show that as the dimension $n$ increases, the
logarithm of the sequence of intrinsic volumes of $\{{\cal S}_n(\sigma,
\rho)\}$ converges to a limit function under an appropriate scaling. The upper
bound on capacity is then expressed in terms of this limit function. We derive
the asymptotic capacity in the low and high noise regime for the $(\sigma,
\rho)$-power constrained AWGN channel, with strengthened results for the
special case of $\sigma = 0$, which is the amplitude constrained AWGN channel.
