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A conjecture of Freiman gives an exact formula for the largest volume of afinite set $A$ of integers with given cardinality $k = |A|$ and doubling $T =|2A|$. The formula is known to hold when $T \le 3k-4$, for some small rangeover $3k-4$ and for families of structured sets called chains. In this paper weextend the formula to sets of every dimension and prove it for sets composed ofthree segments, giving structural results for the extremal case. A weakerextension to sets composed of a bounded number of segments is also discussed.

Additive Volume of Sets Contained in Few Arithmetic Progressions