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For all $$p>1$$ p > 1 and all centrally symmetric convex bodies, $$K\subset {\mathbb {R}}^d$$ K ⊂ R d defined Mf as the centered maximal function associated to K . We show that when $$d=1$$ d = 1 or $$d=2$$ d = 2 , we have $$||Mf||_p\ge (1+\epsilon (p,K))||f||_p$$ | | M f | | p ≥ ( 1 + ϵ ( p , K ) ) | | f | | p . For $$d\ge 3$$ d ≥ 3 , let $$q_0(K)$$ q 0 ( K ) be the infimum value of p for which M has a fixed point. We show that for generic shapes K , we have $$q_0(K)>q_0(B(0,1))$$ q 0 ( K ) > q 0 ( B ( 0 , 1 ) ) .

Lower Bounds and Fixed Points for the Centered Hardy–Littlewood Maximal Operator