We study full-information online linear optimisation on the unit
ℓp ball in
Rd, for fixed
2≤p<∞ and losses of dual norm at most one. Let
ΔT denote the cumulative loss of the best fixed action above each round's minimum. A follow-the-regularised-leader (FTRL) algorithm attains, on every loss sequence and without knowing
T or
ΔT, regret
Op(1+min{ΔT1−1/p,d1/2−1/pΔT}). Matching lower bounds show that this rate is minimax for each fixed
p, and its constant
(2+o(1))p/logp is optimal as
p→∞ among guarantees uniform in
d and
T. The algorithm uses a bounded radial regulariser and switches at most once to its quadratic augmentation. On some paths where follow-the-leader (FTL) has regret one, its regret grows with
T. A switching extension keeps the sharp constant and also guarantees
RegT≤2RegTFTL. For i.i.d. losses whose mean has no zero coordinates, we compute the exact
logT coefficient of the expected regret, which separates FTL's variance term from the bias of the regulariser.