Find the extrema of a linear form on a unit ball

Question

Let $ T: \mathbb R^p \to \mathbb R $ such that $ T(X)= a_1x_1 + a_2x_2 + .....a_px_p$ with ${A_v} = (a_1,a_2,...a_p) \in \mathbb R^p. $

Let $E =\{ X \in \mathbb R^p : \sum_{i=1}^{p} \frac{x_i^2}{\alpha_i^2} \leq 1\}$

1) Find $T(E)$

Using Lagrange Multipliers I have a maximization problem of $a_1x_1/\alpha_1 +a_2x_2/\alpha_2 + ... a_px_p/\alpha_p $ subject to $\sum_{i=1}^{p} \frac{x_i^2}{\alpha_i^2} \leq 1$ From where I obtain that $X_n = \frac{\alpha_n a_n}{2}$

2) Find $x_*$ and $x^*$ such that $ T(x^*) = \sup T$ and $T(x_*) = \inf T$

Answer 1

Using Lagrange multipliers we obtain that $X_n = (\alpha_n*A_n)/2$ From there we obtain that if we take $K= \sqrt(\sum_{i=1}^p \frac{\alpha_i^2 A_i^2}{2}) = \frac{\sqrt(\sum_{i=1}^p{\alpha_i^2 A_i^2}}{2}$

From there we have that T(E) $\subset$ [-K,K] We show the inverse contention by setting $-x_* = x^* = (\frac{A_v}{2\alpha_v^2(4)})$

Using that, we obtain that $T(x_*)= -K ; T(x^*) = K $ Which implies $[-K,K] \subset T(E)$

T(E) = [-K,K]

Answer 2

Hint: do the change of variables $$ x'_i=\frac {x_i}{\alpha_i}$$ $ E $ becomes an honest (symmetric!) unit ball.