grandes-ecoles 2018 Q22

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 $C_{t}$. Show that
$$d(X, C) \leqslant \left\|(1 - \lambda)(Y_{\varepsilon_{n}} + \varepsilon_{n} e_{n}) + \lambda(Y_{-\varepsilon_{n}} - \varepsilon_{n} e_{n}) - X\right\|$$

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 $C_{t}$. Show that

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

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