I want to choose such two norms $\Vert\cdot\Vert_a, \; \Vert\cdot\Vert_b$ for which:
$$\Vert\cdot\Vert_c := \max\{\Vert\cdot\Vert_a, \; \Vert\cdot\Vert_b\}$$
Is a norm inducted by inner product.
My work so far
According to Maximum of two norms is norm I'm asured that my object is well defined. Norm is inducted by inner product when the following formula is satisfied:
$$\Vert x+y \Vert^2 + \Vert x-y \Vert^2 = 2(\Vert x \Vert^2 + \Vert y \Vert^2)$$
Let's define $A := (x_1, y_1)$, $B:= (x_2, y_2)$ and rewrite the formula above:
$$\Vert A+B\Vert_c^2 + \Vert A-B \Vert_c^2 = 2(\Vert A\Vert_c^2 + \Vert B \Vert_c^2)$$
$$\Vert (x_1+x_2, y_1+y_2)\Vert_c^2 + \Vert (x_1-x_2, y_1-y_2)\Vert_c^2 = 2(\Vert (x_1, y_1)\Vert_c^2+\Vert (x_2, y_2)\Vert_c^2)$$
$$ \max{\{\Vert x_1 + x_2 \Vert_a, \Vert y_1 + y_2 \Vert_b\}}^2 + \max{\{\Vert x_1 - x_2 \Vert_a, \Vert y_1 - y_2 \Vert_b\}}^2 = 2( \max{\{\Vert x_1 \Vert_a, \Vert y_1 \Vert_b\}}^2 + \max{\{\Vert x_2 \Vert_a, \Vert y_2 \Vert_b\}}^2)$$
And here I wasnt sure which norms $a, b$ to chose. I tried some combinations $\mathbb{l}_1$ with supremum but it doesn't give me more than just bizzare looking expressions.
Could you please help me findin some norms $a$ and $b$ for which parallelogram law holds ?
Here is a different idea. The unit ball of the $\|\cdot\|_c$ norm is equal to the intersection of the unit balls of the $\|\cdot\|_a$ and $\|\cdot\|_b$ norms. Now choose two norms such that the intersection of their unit balls is the Euclidean unit ball.
Here is one on $\mathbb R^2$: $$ \|(x_1,x_2)\|_a:= \begin{cases} \sqrt{x_1^2 + x_2^2} & \text{if }x_1x_2 \ge 0\\\max(|x_1|,|x_2|) & \text{if }x_1x_2 < 0. \end{cases} $$ Define $\|(x_1,x_2)\|_b:=\|(-x_1,x_2)\|_a$. Since $\max(|x_1|,|x_2|)\le\sqrt{x_1^2 + x_2^2}$, we have $\max(\|(x_1,x_2)\|_a,\|(x_1,x_2)\|_b) = \sqrt{x_1^2+x_2^2}$.