Graphs of convex functions are sigma 1-straight

摘要

A set E subset of or equal to R-n is s-straight for s > 0 if E has finite Method II outer s-measure equal to its Method I outer s-measure. If E is Method II s-measurable, this means E has finite Hausdorff s-measure equal to its Hausdorff s-content. The graph Gamma of a convex function f : [a, b] --> R is shown to be a countable union of 1-straight sets, and to contain a 1-straight set maximal in the sense that its Hausdorff 1-measure equals the diameter of Gamma. [References: 9]