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A linear operator      T : L p ( Ω  ) →  L p ( Ω  1 ) T:{L^p}\left ( \Omega \right ) \to {L^p}\left ( {{\Omega _1}} \right )       is said to be almost isometric on the positive cone (a.i.p.c.) with distortion coefficient      ε  \varepsilon       if there is an      ε  , 0 ≤  ε  ≤  1 \varepsilon ,0 \leq \varepsilon \leq 1      , such that      ( 1 −  ε  ) ‖ f ‖ ≤  ‖ T f ‖ ≤  ‖ f ‖ \left ( {1 - \varepsilon } \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq \left \| f \right \|       for all nonnegative functions      f f      . We prove that if      p ∈  [ 2 , ∞  ) p \in [2,\infty )      , then there are continuous functions      a p ( ⋅  ) {a_p}\left ( \cdot \right )       and      b p ( ⋅  ) {b_p}\left ( \cdot \right )       defined on      [ 0 , 1 ) [0,1)      , with      a p ( 0 ) = b p ( 0 ) = 1 {a_p}\left ( 0 \right ) = {b_p}\left ( 0 \right ) = 1      , so that if      T : L p ( Ω  ) →  L p ( Ω  1 ) T:{L^p}\left ( \Omega \right ) \to {L^p}\left ( {{\Omega _1}} \right )       is an a.i.p.c. operator with distortion coefficient      ε  \varepsilon      , then      a p ( ε  ) ‖ f ‖ ≤  ‖ T f ‖ ≤  b p ( ε  ) ‖ f ‖ {a_p}\left ( \varepsilon \right )\left \| f \right \| \leq \left \| {Tf} \right \| \leq {b_p}\left ( \varepsilon \right )\left \| f \right \|       for all      f ∈  L p ( Ω  ) f \in {L^p}\left ( \Omega \right )      . We also prove that such functions      a p ( ⋅  ) {a_p}\left ( \cdot \right )       and      b p ( ⋅  ) {b_p}\left ( \cdot \right )       exist for      p p       in the range      ( 1 , 2 ) \left ( {1,2} \right )       if, in addition,      T T       is either positive or has dense range.

On operators that are almost isometric on the positive cones of 𝐿^{𝑝}-spaces, 1<𝑝<+∞