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Let $f$ be a convex function. The perspective of its conjugate is defined as $$ h(y,t) = t f^\star(y / t) $$ for $t > 0$ and $y/t \in \operatorname{dom}f^\star$.

Very similarly, if we take a fixed $a > 0$ and compute the convex conjugate of $a f$, we get $$ (a f)^\star(y) = a f^\star( y / a ).$$

Is there any deeper reason why these two formulas are so similar?

gerw
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  • For an introduction to the convex conjugate, see here – Jean Marie Nov 15 '21 at 21:25
  • A good reference here. – Jean Marie Nov 15 '21 at 21:27
  • The operation $(f \lambda)(x) = \lambda f({x \over \lambda})$ for $\lambda>0$ (some details omitted) is referred to as right scalar multiplication by Rockafellar. I would recommend Rockafellar's "Convex Analysis" as an additional reference. – copper.hat Nov 15 '21 at 22:20
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    Really interesting question -- I have not seen something purported as a 'reason' for this. However, it's kinda cool that the theory of perspective functions illuminates that we can allow $a$ to vary while still preserving convexity! Another good ref is Bauschke & Combettes, section 24.6. – Zim Nov 15 '21 at 23:25

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