  We consider a family of heights defined by the $L_p$ norms of polynomials
with respect to the equilibrium measure of a lemniscate for $0 \le p \le
\infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler
measure) and $p=\infty$ corresponds to the standard supremum norm. This special
choice of the measure allows to find an explicit form for the geometric mean of
a polynomial, and estimate it via certain resultant. For lemniscates satisfying
appropriate hypotheses, we establish explicit polynomials of lowest height, and
also show their uniqueness. We discuss relations between the standard results
on the Mahler measure and their analogues for lemniscates that include
generalizations of Kronecker's theorem on algebraic integers in the unit disk,
as well as of Lehmer's conjecture.
