Statement
In 1949 Charles Loewner proved that every metric on the 2-torus satisfies the optimal inequality
where "sys" is its systole, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the Hermite constant in dimension 2, so that Loewner's torus inequality can be rewritten as
The inequality was first mentioned in the literature in Pu (1952).
Read more about this topic: Loewner's Torus Inequality
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