= Solution
For a bounded connected domain with the <interior cone property>, and $1\leq p<n$, the mean-zero form of the <Poincaré inequality> is
$$
\boxed{\|v-v_\Omega\|_{L^p(\Omega)}\leq C_{\Omega,p}\|\nabla v\|_{L^p(\Omega)},\qquad v_\Omega=\frac1{|\Omega|}\int_\Omega v.}
$$
No zero boundary trace is imposed. Subtracting the mean is essential because nonzero constants have zero <gradient>.
Suppose the estimate failed. Subtract means and normalize to obtain $v_j\in W^{1,p}(\Omega)$ with
$$
\int_\Omega v_j=0,\qquad\|v_j\|_p=1,\qquad\|\nabla v_j\|_p\longrightarrow0.
$$
This sequence is bounded in the <Sobolev space> $W^{1,p}$. The allowed <Rellich-Kondrachov compactness theorem> on cone domains provides a subsequence converging strongly in $L^p(\Omega)$ to $v$. Its mean is zero and its <norm> is one. For any compactly supported smooth <test function> $\phi$,
$$
\int v\partial_i\phi=\lim_j\int v_j\partial_i\phi=-\lim_j\int(\partial_i v_j)\phi=0.
$$
Thus every <weak derivative> of $v$ vanishes. To show this implies constancy, mollify on any ball compactly contained in $\Omega$. The resulting smooth function has zero <gradient> and is constant on the smaller ball. Passing to its $L^p$ limit makes $v$ constant almost everywhere on that ball. Overlapping balls have the same constant, and <connectedness> connects any two points by a chain of such balls. Hence $v$ is constant almost everywhere throughout $\Omega$. Its zero mean makes it zero, contradicting <norm> one. This proves the stated <Poincaré inequality>. The frequently used stronger Sobolev-Poincare formulation also follows: with $p^*=np/(n-p)$, the cone-domain <Sobolev embedding theorem> gives $\|w\|_{p^*}\leq C_S(\|w\|_p+\|\nabla w\|_p)$. Insert $w=v-v_\Omega$ and the estimate just proved to obtain
$$
\boxed{\|v-v_\Omega\|_{L^{p^*}(\Omega)}\leq C_S(1+C_{\Omega,p})\|\nabla v\|_{L^p(\Omega)}.}
$$
Thus both the same-exponent and critical-exponent mean-zero versions are covered. Connectedness cannot be dropped without subtracting a separate mean on every component. The usual meaning of the stated range is $1\leq p<n$; $W^{1,p}$ with $p<1$ is not part of this Sobolev-space formulation.
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