    \begin{align*}
        A(t\cdot (x_1, y_1) + (1-t) \cdot (x_2, y_2)) &= (tx_1 + (1-t)x_2)^c (ty_1 + (1-t)y_2) - x_0^c y_0\\
        &> (x_1^t x_2^{1-t})^c (y_1^t y_2^{1-t}) - x_0^c y_0\\
        &= (x_1^c y_1)^t (x_2^c y_2)^{1-t} - x_0^c y_0\\
        &\ge (b + x_0^c y_0)^t (b + x_0^c y_0)^{1-t} - x_0^c y_0\\
        &= b.
    \end{align*}
