Barbell versus bullet

Use continuous compounding, so a zero maturing at $T_i$ has price $e^{-yT_i}$. (a) Show that a portfolio of zeros with value weights $w_i$ has duration $D=\sum w_iT_i$ and convexity $C=\sum w_iT_i^2$. (b) Prove that a barbell (zeros at $T_1<T_2$) with the same duration as a bullet (one zero at $T_0=D$) has strictly greater convexity, and that the excess equals the value-weighted variance of maturities. (c) Compute both convexities for a 2y/10y barbell matched to a 6y bullet. (d) Under a parallel shift, why can the barbell not be dominated? What kind of curve move hurts the barbell relative to the bullet?