As stated, the answer is no, Freiman's inequality no longer holds. The counter example is $A$ being the vertices of an equilateral triangle on the unit circle. I found this by looking at the proof of Freiman's inequality (given as Lemma 5.13 of Tao-Vu book on Additive Combinatorics) and see where the proof fails.

In my particular setting, the set $A$ satisfies some additional convexity conditions, so perhaps the result may hold for such situations.

UPDATE: it turned out that the proposed inequality held under one extra assumption: that the origin is out side the convex hull of $A$. That will rule out the examples such as above, and still implies the classical inequality. The proof follows the same line as the original proof, with some extra care needed.

As stated, the answer is no, Freiman's inequality no longer holds. The counter example is $A$ being the vertices of an equilateral triangle on the unit circle. I found this by looking at the proof of Freiman's inequality (given as Lemma 5.13 of Tao-Vu book on Additive Combinatorics) and see where the proof fails.

In my particular setting, the set $A$ satisfies some additional convexity conditions, so perhaps the result may hold for such situations.

UPDATE: it turned out that the proposed inequality held under one extra assumption: that the origin is outside the convex hull of $A$. That will rule out examples such as above, and still implies the classical inequality. The proof follows the same line as the original proof, with some extra care needed.

As stated, the answer is no, Freiman's inequality no longer holds. The counter example is $A$ being the vertices of an equilateral triangle on the unit circle. I found this by looking at the proof of Freiman's inequality (given as Lemma 5.13 of Tao-Vu book on Additive Combinatorics) and see where the proof fails.

In my particular setting, the set $A$ satisfies some additional convexity conditions, so perhaps the result may hold for such situations.

As stated, the answer is no, Freiman's inequality no longer holds. The counter example is $A$ being the vertices of an equilateral triangle on the unit circle. I found this by looking at the proof of Freiman's inequality (given as Lemma 5.13 of Tao-Vu book on Additive Combinatorics) and see where the proof fails.

In my particular setting, the set $A$ satisfies some additional convexity conditions, so perhaps the result may hold for such situations.

UPDATE: it turned out that the proposed inequality held under one extra assumption: that the origin is out side the convex hull of $A$. That will rule out the examples such as above, and still implies the classical inequality. The proof follows the same line as the original proof, with some extra care needed.