# Linear interpolation of local vol no arbitrage

We already know the equivalence between local vol, implied vol and option price and there are one-one maps between pairs: $$(\sigma_{local},K),\ (\sigma_{implid},K),\ (C,K)$$ here $$K$$ is the strike and $$C$$ is the call price. And the one-one maps are determined by Black-Scholes formula.

And we know that one of the no-arbitrage condition is that call price $$C$$ should be a convex function against the strike $$K.$$

So, given some market equivalent sample points $$(\sigma^i_{local},K^i),\ (\sigma^i_{implid},K^i),\ (C^i,K^i)$$ assume they meet the no arbitrage condition as precondition, then we know that linear interpolation of implied vol $$\sigma^i_{implid}$$ is not allowed.

My question is whether linear interpolation of local vol $$\sigma^i_{local}$$ is allowed? Since linear interpolation of market price $$C^i$$ seems allowed.

If linear interpolation of local vol $$\sigma^i_{local}$$ is allowed, what's the reason? It seems related to the convexity of the maps between $$(\sigma^i_{local},K^i)\ (C^i,K^i)$$ and $$(\sigma^i_{implid},K^i),\ (C^i,K^i).$$