Friends any hint as to why is this set of equations a test of linearity of a derivative security?
From Taleb - Dynamic Hedging pg. 11
,, Derivatives are not always linear, convex, or concave across all moves (See Figures 1.2A-D, I did put a picture excerpt from the book down ). A test of local linearity of a derivative security (that is a function of the underlying asset) between asset prices $S_1$ and $S_2$ with $0<λ<1$, will satisfy the following equality: $$V(λS_1 + (1-λ)S_2) = λV(S_1) + (1 - λ)V(S_2)$$
It will be convex between $S_1$ and $S_2$ if: $$V(λS_1 + (1-λ)S_2) ≤ λV(S_1) + (1 - λ)V(S_2)$$ It will be concave if: $$V(λS_1 + (1-λ)S_2) ≥ λV(S_1) + (1 - λ)V(S_2)$$
Any hint which mathematical theorem is behind this equation. $V$ i assume means Value of the derivative security. Is there somewhere to read to hone some insight about this for the unenlightened scholar?