Along with their legal brief, attorneys filed a baseball shoe in federal court in New York City last week and claimed it proves that Nike Inc., the nation's biggest sports shoemaker, is trying to monopolize the market illegally.

Allegedly borrowed from a player in the World Series, the shoe has a "Nike" label on the heel and the familiar Nike "swoosh stripe" on the side, but molded into the sole is the name "Brooks" -- Nike's chief competitor in the $470-million-a-year running shoe business.

Nike pasted its own stripes on other people's shoes to placate professional athletes who are paid to wear Nikes but secretly prefer another brand, Brooks employe Eric Maitland contends in an affadavit field along with the shoe.

The shoe is exhibit No. 1 in an antitrust suit filed against Nike by Brooks Shoe Manufacturing Co. Inc. of Hanover, Pa.

The lawsuit claims Brooks has suffered $10 million in damages because of illegal competition by Nike. If the courts uphold the claim, Brooks will be eligible for triple damages -- $30 million.

The Brooks lawsuit is the company's response to a $25 million antitrust case that Nike field last February against the Rubber Manufacturers Association, a trade association, and seven shoemakers, including Brooks.

Nike contends it is the victim of a conspiracy by the other shoemakers, who have urged the U.S. Customs Service to slap extra import levies on foreign-made running shoes.

Nike is based in Beaverton, Ore., and had shoe factories in Maine and New Hampshire, but the company imports 80 percent of its shoes from the Far East and Europe.

Nike makes eight of the 10 best-selling running shoe models and controls more than 55 percent of the $470-million-a-year sport shoe business, the lawsuits claim.

Keeping pace with the race in running, Nike's shoe business has sprinted from $29 million a year in 1977 to $270 million last year.

Brooks contends Nike has run up the huge lead in the footwear race through a series of allegedly illegal tactics, including industrial espionage, payoff to athletes and unfair competitive practices.

Nike accuses Brooks of cultivating a favored relationship with a magazine called Runner's World to win top ratings in the publication's annual tests of running shoes. In the ratings just published, Brooks shoes won three of the top five places for women's training shoes and four of the top 10 rated men's training shoes.

Refusing to participate in the tests, Nike complained that the publisher of Runner's World, Bob Anderson, also owns a company that sells Brooks shoes. Brooks contributed funds to cosponser races with Runner's World, the Nike makers charged.

Nike pulled its advertising out of Runner's World and recently acquired its own magazine, called Running.

Brooks says the new magazine is part of Nike's illegal effort to dominate the running shoe market, charging that Nike tried to discredit Runner's World because its shoes got poor marks in the magazine's ratings. Nike also opened its own running shoe stores in a further effort to eliminate competition, Brooks complained.