  In this paper, we explore carrot John domains within variational problems,
dividing our examination into two distinct sections. The initial part is
dedicated to establishing the lower semicontinuity of the (optimal) John
constant concerning Hausdorff convergence for bounded John domains. This result
holds promising implications for both shape optimization problems and
Techm\"uller theory.
  In the subsequent section, we demonstrate that an unbounded open set
satisfying the carrot John condition with a center at $\infty$, appearing in
the Mumford-Shah problem, can be covered by a uniformly finite number of
unbounded John domains (defined conventionally through cigars). These domains,
in particular, support Sobolev-Poincar\'e inequalities.
