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RobPratt
RobPratt
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dohmatob
dohmatob
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Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

  • If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
  • If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |a|\min(1,R/\sqrt c)$$\alpha = |b|\min(1,R/\sqrt c)$.
  • $\lim_{c \to \infty} \alpha = |b|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

  • If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
  • If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |a|\min(1,R/\sqrt c)$.
  • $\lim_{c \to \infty} \alpha = |b|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

  • If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
  • If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |b|\min(1,R/\sqrt c)$.
  • $\lim_{c \to \infty} \alpha = |b|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.
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dohmatob
dohmatob
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Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^d \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$$C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

  • If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
  • If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |a|\min(1,R/\sqrt c)$.
  • $\lim_{c \to \infty} \alpha = |a|R'$$\lim_{c \to \infty} \alpha = |b|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^d \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

  • If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
  • If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |a|\min(1,R/\sqrt c)$.
  • $\lim_{c \to \infty} \alpha = |a|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

  • If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
  • If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |a|\min(1,R/\sqrt c)$.
  • $\lim_{c \to \infty} \alpha = |b|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.
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dohmatob
dohmatob
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dohmatob
dohmatob
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added 93 characters in body
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dohmatob
dohmatob
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dohmatob
dohmatob
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