Is a cylinder a Lipschitz domain?

I’m wondering if the domain $(0,T)\times \Omega$ is a Lipschitz domain ($T$ is a positive real number), provided that $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with Lipschitz boundary, and how to prove or disprove this fact.

Thank you

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Yes, it is. There are three parts of the boundary to check.

• $(0,T)\times \partial \Omega$. Locally, and in a right coordinate system, $\Omega$ is the set above the graph of some Lipschitz function. Multiplying it by a line segment maintains this property: the new Lipschitz function can be taken to be independent of $t$.

• $\{0,T\}\times \partial \Omega$. Begin as above, and then slightly tilt the coordinate system so that its axes are not parallel to the hyperplane $t=0$. Then the plane $t=0$ is also the graph of a Lipschitz function. The product domain lies above the graph of the maximum of these two Lipschitz functions: one linear, one coming from $\partial \Omega$.

• $\{0,T\}\times \Omega$ is flat.