# Wrong understanding of implied vol cannot be linear interpolated

There is a well known conclusion that implied vol cannot be linear interpolated. But I am not sure this interpolation is for strike $$K$$ or time $$T$$?

That's definitely true for time. When strike $$K$$ is fixed, interpolation of implied vols respect to time $$T$$ can make the Calendar arbitrage. Surely we assume the interpolation points satisfy the no arbitrage condition.

But is it true when time $$T$$ is fixed, interpolation of implied vols respect to strike $$K$$ can make a arbitrage.

I really struggled for related problems, can anyone give me a precise answer?