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Michael Hardy
Michael Hardy
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Consider for $r,c>0$ the set $$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$$$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_{r,c}} ||x||_2 = 0.$$\inf_{x \in X_{r,c}} \|x\|_2 = 0.$ But is it possible to compute $$s_{r,c}:=\sup_{x \in X_{r,c}} ||x||_2 \quad ?$$$$s_{r,c}:=\sup_{x \in X_{r,c}} \|x\|_2 \text{ ?}$$ Does at least $\lim_{c \to 0} s_{r,c}=r$ hold?

Consider for $r,c>0$ the set $$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_{r,c}} ||x||_2 = 0.$ But is it possible to compute $$s_{r,c}:=\sup_{x \in X_{r,c}} ||x||_2 \quad ?$$ Does at least $\lim_{c \to 0} s_{r,c}=r$ hold?

Consider for $r,c>0$ the set $$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_{r,c}} \|x\|_2 = 0.$ But is it possible to compute $$s_{r,c}:=\sup_{x \in X_{r,c}} \|x\|_2 \text{ ?}$$ Does at least $\lim_{c \to 0} s_{r,c}=r$ hold?

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Iosif Pinelis
Iosif Pinelis
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Consider for $r,c>0$ the set $$X_c=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$$$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_c} ||x||_2 = 0.$$\inf_{x \in X_{r,c}} ||x||_2 = 0.$ But is it possible to compute $$s_c:=\sup_{x \in X_c} ||x||_2 \quad ?$$$$s_{r,c}:=\sup_{x \in X_{r,c}} ||x||_2 \quad ?$$ Does at least $\lim_{c \to 0} s_c=r$$\lim_{c \to 0} s_{r,c}=r$ hold?

Consider for $r,c>0$ the set $$X_c=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_c} ||x||_2 = 0.$ But is it possible to compute $$s_c:=\sup_{x \in X_c} ||x||_2 \quad ?$$ Does at least $\lim_{c \to 0} s_c=r$ hold?

Consider for $r,c>0$ the set $$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_{r,c}} ||x||_2 = 0.$ But is it possible to compute $$s_{r,c}:=\sup_{x \in X_{r,c}} ||x||_2 \quad ?$$ Does at least $\lim_{c \to 0} s_{r,c}=r$ hold?

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SequenceGuy
SequenceGuy
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Maximization of $\ell^2$-norm

Consider for $r,c>0$ the set $$X_c=\{x \in \ell^1(\mathbb{N}) \mid ||x||_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_c} ||x||_2 = 0.$ But is it possible to compute $$s_c:=\sup_{x \in X_c} ||x||_2 \quad ?$$ Does at least $\lim_{c \to 0} s_c=r$ hold?