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In this paper, we show how geometry plays in the study of the Furstenbergconjecture (refer to~\cite{F}). Let $p>1$ and $q>1$ be two relative primepositive integers. We prove that a non-atomic $p$- and $q$-invariant measurehaving balanced geometry must be the Lebesgue measure. In the proof, we willnot assume the ergodicity of the measure. The result provides an intuitivegeometric criterion to either prove the Furstenberg conjecture or construct acounter-example.

Geometry in the Furstenberg Conjecture