We observe Euro-call option prices at discrete strikes, say, $C_1>C_2>C_3...$ at several strikes $K_1<K_2<K_3,...$. (The maturity is the same)
It is well-known that the option prices are convex with repsect to $K$. Hence, it is easy to see that, we can just link these points to have a piecewise linear prices line w.r.t. $K$ as an upper bound the prices between $K_i$ and $K_{i+1}$.
However, what about the lower bounds? It there any simple lower bound for the prices between $K_i$ and $K_{i+1}$?
Due to the convexity, it seems that setting the price between $K_i$ and $K_{i+1}$ equal to $C_i$ could violate the convexity.