13s are generally heavier, the wheel/tyre combo is quite a chunk heavier than a little 10
they are also "generally" or "Normally" outset compared to the usual 10 or 12" mini rims, causeing a fair bit of load to both wheel bearings & steering,,, not to mention the very poor scrub radius created by such an outset rim
lots of sports-sedan minis run 13 x8/9/or 10s & usually in circuit racing it`s not too much of an issue at high speeds & sweeping corners, however it`s still an issue,,, but the grip level gained seems to outweigh the scrub-radius & weight problems in such circumstance,,, in other words lots of people would have simply "put-up-with" & driven around the issues
however,,, the humble little 10" wheel/tyre combo with "decent" sticky tyres is hard to beat for tight corners & stop-start type action,,, quick respons & nimble handling are a pleasure to drive & quite fun
sure there`s less grip but you have to make you`re own mind up depending on a number of things
the power you have,,, if you have a burnt valved 850 then 13x10s with make it like a 2cyl piece of crap in the power dept, as if it isn`t bad enough with a burnt valved 850 in a hillclimb
but on the other hand,,, if you had a twin-cam turbo injected rocket ship with 300hp,,, then maybe 13x10s with super sticky rubber will be your best friend & just put up with the slight problem of scrub radius & weight & suspension/wheel brg & steering loads etc
so,,, think about the physics of your combination-->,,, your "package" ... & decide if you want nimble fun dancing easy light chuck-ability with your "medium" (whatever) power you currently have...
or do you like the tough look & stance & aggressiveness of the 13s & maybe you think you have awesome wicked power that little pissy 10s just won`t cope,,, then maybe 13s are the go
don`t forget tho,,, if you run 13X10s (or something similar) then most times the rolling circumfrence is longer & therefore gearing becomes an issue,,, it`s a bit hard to get a pissy thin little 15tooth diff pinion to cope with massive 250hp & 13x10" sticky slicks,,, & also the height of the car is raised with the bigger wheel/tyre combo
so then you start changing suspension control arm "pick-up" points on the subframe , raising the inner mounting point & shortening towers & hi-los & radically changing geometry of all your arms & steering rack positions to try to correct it all to suit the higher stance,,, blah blah blah,,, it goes on & on
so if you want to just drive your mini at hillclimbs & have fun?... or do you want the fastest mini in the world & spend the next 5 years perfecting an aggressive over-powered monster?
hope you get my point of all the variables & work required to get a 13" wheel`d car "closer" to being a great handling car

or enjoy the easy & fun & cheapness & simple-ness of a 10" thing
