Is convex optimisation in P?
In 1998, Michel X. Goemans gave an ICM talk, in which, he addressed this issue:"Semidefinite programs can be solved(or more precisely, approximated) in polynomial-time within any specific accuracy either by the ellipsoid algorithm or more efficiently through interior-point algorithms...The above algorithms produce a strictly feasible solution(or slightly infeasible for some versions of the ellipsoid algorithm) and, in fact, the problem of deciding whether a semidefinite program is feasible(exactly) is still open. A special case of semidefinite programming feasibility is the square-root-sum problem. The complexity of this problem is still open."
http://garden.irmacs.sfu.ca/op/complexity_of_square_root_sum

In 1976, Ron Graham, Michael Garey, and David Johnson could not show some geometric optimization problems such as whether Euclidean Traveling Salesman Problem is NP-complete(they can only show the problem is NP-hard), the reason is that they could not show whether the square-root-sum problem is polynomial time solvable or not. 
https://rjlipton.wordpress.com/2009/03/04/ron-graham-gives-a-talk/

The square-root-sum problem is a long open problem which puzzles scholars from computational geometry, optimization, computational complexity, game theory and some other areas a lot as they all at some point, figure out the main obstacle for their problems is to handle the square-root-sum problem. 

The most remarkable progress towards this problem is by Eric Allender and his co-authors, in 2003, they showed this problem lies in the 4th level of the Counting Hierarchy. 
http://ftp.cs.rutgers.edu/pub/allender/slp.pdf

So based on above facts, one can't solve convex optimization problem in (true) polynomial time with the Ellipsoid method and Interior Point method. 

The big O notation is to measure the running time of the algorithm in worst case. However, in practice, the worst case may be a very rare event, that's why you can not use it to measure the practical performance.