In what case did the US Supreme Court establish the principle of 'one-man-one-vote'?

Answer 1

Reynolds v. Sims, 377 US 533 (1964)

The "one man, one vote" rule (also called "one person, one vote") derives from the US Supreme Court ruling in Reynolds v. Sims, 377 US 533 (1964) that held state political districts of unequal size resulted in under-representation of some citizens' interests and over-representation of others'. This was considered "unrepublican," per Article IV, Section 4 of the Constitution, and also unconstitutional under the Fourteenth Amendment Equal Protection Clause. In order to meet constitutional standards, districts had to be reapportioned so each had approximately equal population.

Wesberry v. Sanders, 376 US 1 (1964) applied the same principle to districts of the US House of Representatives.

Both Wesberry and Reynolds decisions were predicated on the landmark ruling in Baker v. Carr, 369 US 186 (1962), in which the US Supreme Court decided reapportionment of state legislative districts was not a "political question" that should be resolved through legislation. The Court found legislative conflicts of interest raised justiciable issues that could be addressed and resolved by the Federal courts.