grandes-ecoles 2018 Q23

grandes-ecoles · France · centrale-maths1__official Vectors Introduction & 2D Inequality or Proof Involving Vectors

Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$. For $t$ in $\{-1, 1\}$, $Y_{t}$ denotes the projection of $X'$ onto $C_{t}$. Deduce that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + \left\|(1 - \lambda)(Y_{\varepsilon_{n}} - X') + \lambda(Y_{-\varepsilon_{n}} - X')\right\|^{2}$$
then that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda)\|Y_{\varepsilon_{n}} - X'\|^{2} + \lambda\|Y_{-\varepsilon_{n}} - X'\|^{2}$$
Thus, show the inequality
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda)d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$

Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$. For $t$ in $\{-1, 1\}$, $Y_{t}$ denotes the projection of $X'$ onto $C_{t}$. Deduce that

$$d(X, C)^{2} \leqslant 4\lambda^{2} + \left\|(1 - \lambda)(Y_{\varepsilon_{n}} - X') + \lambda(Y_{-\varepsilon_{n}} - X')\right\|^{2}$$

then that

$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda)\|Y_{\varepsilon_{n}} - X'\|^{2} + \lambda\|Y_{-\varepsilon_{n}} - X'\|^{2}$$

Thus, show the inequality

$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda)d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$

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