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We compute the exact lower bounds for some averages arising in ergodic theory. In particular, we prove that for any measure-preserving system (X,\mathcal{B},\mu,T) with \mu(X)<\infty, any A\in\mathcal{B} and any N\in\mathbb{N}, N^{-1}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}A)\geq\sqrt{\mu(A)^{2}+(\mu(X)-\mu(A))^{2}}+\mu(A)-\mu(X).
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