piecewise linear infimal convolution

data = {0, 10, 0, 20, 10, 30, 0};
breaks = MapIndexed[
   With[{pos = First[#2], y1 = First[#], 
      y2 = Last[#]}, {y1 (1 + pos - x) + y2 (x - pos), 
      pos <= x < pos + 1}] &, Partition[data, 2, 1]];
f[x_] := Piecewise[breaks];
Plot[f[x], {x, 1, Length[data]}]
h[lambda_] := 
 lambda + Max[
   Table[MaxValue[{breaks[[i]][[1]] - x, 
      Inequality[breaks[[i]][[2]][[1]], breaks[[i]][[2]][[2]], 
       breaks[[i]][[2]][[3]], breaks[[i]][[2]][[4]], 
       Min[breaks[[i]][[2]][[5]], lambda]]}, x], {i, 1, 
     Length[breaks]}]]
Plot[h[zzz], {zzz, 1, Length[breaks]}]