On two new kinds of restricted sumsets

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n\not=a_1\}$$ recently introduced by the second author, when $G$ is the additive group of a field we obtain lower bounds for $|L(A_1,\ldots,A_n)|$ and $|C(A_1,\ldots,A_n)|$ via the polynomial method. Moreover, when $G$ is torsion-free and $A_1=\cdots=A_n$, we determine completely when $|L(A_1,\ldots,A_n)|$ or $|C(A_1,\ldots,A_n)|$ attains its lower bound.