How do you explain to laymen the fastest way around a corner using maths?

When I was a teenager one of my friends at the time was convinced the fastest way around a racetrack was to hug the inside of the corner. No amount of me trying to explain the correct strategy of starting wide, turning in, cutting the apex and accelerating out of the corner ending up wide again could convince him otherwise. Even watching an actual race or playing a simulation racing game and trying it himself could not convince him he was wrong.

I'm guessing in this case there would never be any hope to convince this particular person but that recently got me wondering... Is there a way to show using maths what the correct method of cornering is?

I'm thinking maybe it would be something to do with deceleration and acceleration vs distance but the fact is while I'm a great racer in games I couldn't do a complex maths problem if my life depended on it.

So is there anyone here that knows a little about maths and racing that could give me some ideas of how this explanation would go?

Here is a solid explanation, and compares the apex line with the inside of the track and the outside.
http://phors.locost7.info/phors05.htm
Its an idealised scenario but should satisfy what you're looking for. It's all about aiming for the highest average speed through a corner.

But if you're trying to convince someone of this who has watched what pro racing drivers do and still believes otherwise, put him in the bin cause he is a lost cause.

I would hit them in the head with the corner of a calculator until it got through their thick skull. :D

I got part way through the article, and will read the rest later.

I think the trick to the question is 'real life'

If you are travelling at a constant speed and you are not effected by the need to maintain a coefficient of friction with the road surface, the fastest way around a corner is the shortest distance. no questions asked. imagine that track had two cars on it that were travelling at a steady 10km an hour, neither car will skid in the turn but if one is on the outside of the track and one is on the inside of the track, the car on the inside has less distance to travel and will finish the turn faster. it's the same reason why they have staggered starts in the running races in the Olympics.

if however you are driving a car around a corner as fast as you can and have maximum speeds you can travel because you must maintain solid contact with the road at all times, the best path is essentially as straight a line as you can manage through the corner.
The more you try to turn the car the more force is applied to the tyres in a sideways direction, eventually if this force gets too high the tyres will skid. This is related to the coefficient of friction of the tyres against the specific road surface, speed, direction and weight of the car.

The simplest description is, you have to slow down for a corner right? So you make that corner as straight as you can.

Friction does not hold your car to the surface of the road. The ability of the tyre to key into the surface is grip.

The shortest way isn't the fastest way. Doing what is essentially a U turn by hugging the inside is the slowest way possible. The fastest way through a corner is the straightest way with minimal turning. The straightest way is to go wide, then hit the apex and exit wide. This is something you've already even demonstrated to him. I'd just not bother if he is still going to argue it.

Remember the racetrack "Silence" on F-Zero-X?

That track answered this question pretty well ;)