  We introduce and study the communication complexity of computing the inner
product of two vectors, where the input is restricted w.r.t. a norm $N$ on the
space $\mathbb{R}^n$. Here, Alice and Bob hold two vectors $v,u$ such that
$\|v\|_N\le 1$ and $\|u\|_{N^*}\le 1$, where $N^*$ is the dual norm. They want
to compute their inner product $\langle v,u \rangle$ up to an $\varepsilon$
additive term. The problem is denoted by $\mathrm{IP}_N$.
  We systematically study $\mathrm{IP}_N$, showing the following results:
  - For any symmetric norm $N$, given $\|v\|_N\le 1$ and $\|u\|_{N^*}\le 1$
there is a randomized protocol for $\mathrm{IP}_N$ using
$\tilde{\mathcal{O}}(\varepsilon^{-6} \log n)$ bits -- we will denote this by
$\mathcal{R}_{\varepsilon,1/3}(\mathrm{IP}_{N}) \leq
\tilde{\mathcal{O}}(\varepsilon^{-6} \log n)$.
  - One way communication complexity
$\overrightarrow{\mathcal{R}}(\mathrm{IP}_{\ell_p})\leq\mathcal{O}(\varepsilon^{-\max(2,p)}\cdot
\log\frac n\varepsilon)$, and a nearly matching lower bound
$\overrightarrow{\mathcal{R}}(\mathrm{IP}_{\ell_p}) \geq
\Omega(\varepsilon^{-\max(2,p)})$ for $\varepsilon^{-\max(2,p)} \ll n$.
  - One way communication complexity $\overrightarrow{\mathcal{R}}(N)$ for a
symmetric norm $N$ is governed by embeddings $\ell_\infty^k$ into $N$.
Specifically, while a small distortion embedding easily implies a lower bound
$\Omega(k)$, we show that, conversely, non-existence of such an embedding
implies protocol with communication $k^{\mathcal{O}(\log \log k)} \log^2 n$.
  - For arbitrary origin symmetric convex polytope $P$, we show
$\mathcal{R}(\mathrm{IP}_{N}) \le\mathcal{O}(\varepsilon^{-2} \log
\mathrm{xc}(P))$, where $N$ is the unique norm for which $P$ is a unit ball,
and $\mathrm{xc}(P)$ is the extension complexity of $P$.
