Denmark’s Mikkel Hansen takes a shot in the first half against Tunisia on Tuesday. (Elsa/Getty Images)

Athletically speaking, America’s most abundant resource is mesomorphs who run fast, jump high and throw balls at a target or to each other. The sport of team handball requires its competitors to run fast, jump high and throw balls at a target or to each other. It may be hopelessly ignorant and wildly jingoistic to say this, but after spending a day at Future Arena, it is impossible to leave without thinking it: The United States should be awesome at this.

The United States is very far from awesome, having failed to qualify for these Olympics, just as it failed to qualify for the previous four. Actually, the United States failed to qualify for the qualifying tournament. It needed to beat Uruguay to reach the Pan-American Games, the winner and runner-up of which gained passage to Rio. It did not.

At many levels, it makes no sense. America has the right kind of athletes with the right kind of sports backgrounds. True, they have no incentives to play handball, but disregard that for a moment: Really, how long would it take LeBron James to become the best handball player in the world?

“Maybe six months,” U.S. national team Coach Javier Garcia-Cuesta said. “This is just a hypothetical. He has everything. When you see him playing, your mouth drops.”

Denmark’s Mikkel Hansen is pretty much the LeBron James of handball. He was named the International Handball Federation’s world player of the year for the second time last year. This year, he won the equivalent of the MVP in the French league. He would be the right man to ask: Could James be as good as him?

“It’s difficult to talk about that,” Hansen said, laughing. “I admire LeBron James very much. I’m a big basketball fan. The way he sees the court, his vision for the game, is very impressive. There you would have a good start. And physically, he is amazing. But you also need to throw the ball. So, yeah. Maybe.”

The Americans last played handball in the Olympics in 1996, when they qualified as the host nation, finished ninth and got outscored by 30 goals. The men’s team is 38th in the International Handball Federation rankings, just behind Morocco, Australia and Lithuania. The United States is a non-entity in a sport that combines all of the qualities American athletes usually excel at.

“We always talk about it: If the Americans really wanted to play handball, they would be amazing,” Denmark center back Morten Olsen said. “They have so many good athletes, and really big, strong athletes. It would be a big problem. Playing against LeBron James, that would be hard.”

It’s hard to figure why handball is basically invisible in the United States. Unfamiliarity is probably the only explanation. Handball refutes every standard American complaint about soccer. It’s easier to watch on television than hockey. A match takes about two hours less than a typical baseball game. It’s the kind of game kids make up in their backyard and play until darkness falls. There’s contact, excitement and a ball that is thrown rather than kicked. No stick or helmet is required. The rules are easy to learn.

And yet, USA Handball is in such a deep hole it’s hard to envision a way out. It can’t attract the country’s best athletes, so it can’t compete against the best international teams, so it can’t gain popularity, so it can’t entice sponsors, so it can’t raise money to play in many European tournaments, so it can’t gain experience against top-level competition, so it can’t raise awareness, so it can’t attract the country’s best athletes. Few high schools and colleges offer handball, and when’s the last time you saw it on television?

“The sport is perfect for the type of athletes and the type of culture we have in the country,” Garcia-Cuesta said. “The problem is, people don’t know the game and you don’t get the best athletes.”

But what if the best athletes played handball?

“You’re talking about another sport with running and jumping, changing direction and throwing a ball into a net?” U.S. men’s basketball team guard Kyle Lowry said. “Yeah, I think we could figure it out. How much do they make?”

Some of the players in Rio — the ones who actually are the best players in the world — viewed that with skepticism. They said handball is more physical than basketball, though it wasn’t clear where any of them had watched an NBA game up close. The strategies and footwork in handball are more complex than meets the eye.

But still. Say the U.S. men’s basketball players wandered into Future Arena. Their athleticism would be enough to compete, right?

“It would not be a clear win,” Gemany’s Finn Lemke said. “But we would win.”

French player Valentin Porte was not as generous. He predicted the U.S. basketball team would lose, 42-10, to an Olympic handball squad. For perspective, the largest margin of victory through two days of the 12-team Olympic competition has been 15 goals.

“Physically, there would be no problem,” Porte said. “But because of finer points, they would be destroyed. Sorry.”

“They would have problems shooting the ball,” Hansen said. “They would not score that much. It’s much more difficult than you think.”

Garcia-Cuesta is trying to recruit college basketball and football players who fall short of professional leagues. Even that is difficult, because USA Handball lacks the resources or opportunities to entice them. Even the country’s best club teams, Cuesta said, practice just three times a week. The only leagues are regional.

If the United States could pick its handball team from among its best athletes, though, Garcia-Cuesta believes it could win a gold medal by Tokyo 2020.

“That would be an unbelievable team,” Cuesta said. “At that moment, we would all wake up and we would be very happy.”

In handball, teams of six field players, plus a goalie, play with the objective of throwing a one-pound ball — bigger than a softball, smaller than a basketball — into a net.

There’s a restricted zone in front of each net, like an enlarged hockey crease, where only the goalie can enter. Even defenders have to stay out; thus, the predominant defensive alignment is pretty much a semicircular wall.

The most effective offensive move, then, is the jump throw, often made after a series of weaves and picks creates space. Players can leap into the restricted zone as long as they shot before landing. The collisions when players charge through the defensive line make block-charge calls look tame.

The players are bigger than you expect. Nine of Denmark’s 14 players are at least 6 feet 5, and there aren’t many small, quick guys. Several players said the toughest adjustment for a skilled American athlete would be the physicality.

In the past two days, Garcia-Cuesta has received 13 emails from former college athletes interested in handball as a path to the Olympics. He plans to hold a tryout at USA Handball’s headquarters in Auburn, Ala., on Oct. 15. And if Americans in Rio show interest?

“Give them my contact,” Cuesta said.

Jerry Brewer contributed to this report from Rio de Janeiro.