MathDB

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous function and let

g : \mathbb{R} \to \mathbb{R}

be arbitrary. Suppose that the Minkowski sum of the graph of

f

and the graph of

g

(i.e., the set

\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}

) has Lebesgue measure zero. Does it follow then that the function

f

is of the form

f(x) = ax + b

with suitable constants

a, b \in \mathbb{R}

Problem Statement