Let $F\subset \mathbb{R}^n$ be a finite set and $\sigma$ be uniformly distributed over $\{-1,1\}^n$. The usual Rademacher average of $F$ (modulo normalizing factors) is $$ R_n(F)=\mathbb{E}_\sigma \max_{f\in F}\sum_{i=1}^n \sigma_if_i. $$ Now let us define two operations on $F$: $\mathrm{conv}(F)$ and $[F]_\vee$. The former is just the convex hull of the vector-set $F$ in $\mathbb{R}^n$. The latter is defined by $$ [F]_\vee=\{ f\vee g :f,g \in F\}, $$ where $(f\vee g)_i=\max\{f_i,g_i\}$ is the coordinate-wise maximum. It is well-known (and easy to show) that $ R_n( \mathrm{conv}(F)) = R_n( F) $.

Question: is it true that $$ R_n( [\mathrm{conv}(F)]_\vee) = R_n( [F]_\vee) $$ ?

The inequality $ R_n( [F]_\vee) \le R_n( [\mathrm{conv}(F)]_\vee) $ holds due to set containment. Also, $\mathrm{conv}(F)$ is an infinite set, so the $\max$ in the definition should be replaced by a $\sup$.

Update. Fedor Petrov has constructed a counterexample, which I've accepted. The more general conjecture I had was as follows. Define the $k$-fold max operator $[F]_k$ by $$ [F]_k=\{ f_1\vee f_2\ldots\vee f_k :f_i \in F\}. $$ Is there a universal constant $c$ (independent of $n$ and $k$) such that $$ R_n( [\mathrm{conv}(F)]_k) \le c R_n( [F]_k) $$?

  • $\begingroup$ But $[\mathrm{conv}(F)]_\vee$ is not already a finite set? And it looks that the maximum over a subset may be only less than a maximum over a large set, so the opposite inequality should take place. $\endgroup$ – Fedor Petrov Apr 30 at 19:53
  • $\begingroup$ @FedorPetrov Please see edits. $\endgroup$ – Aryeh Kontorovich Apr 30 at 20:58
  • $\begingroup$ so your question is equivalent to "is it true that $R_n( [\mathrm{conv}(F)]_\vee)=R_n( [F]_\vee)$"? $\endgroup$ – Fedor Petrov Apr 30 at 21:02
  • $\begingroup$ yes, I edited again $\endgroup$ – Aryeh Kontorovich Apr 30 at 21:11

It looks that no. Take $n=4$ and $F$ containing four vectors: $f=(2,-2,1,-5); g=(-2,2,1,-5); h=(0,0,-5,1)$. We have $f\vee g=(2,2,1,-5)$, $f\vee h=(2,0,1,1)$, $g\vee h=(0,2,1,1)$. Thus $\max_{w\in [F]_\vee} (w_3+w_4-w_1-w_2)=0$. On the other hand $\frac{f+g}2\vee h=(0,0,1,1)$, therefore $\max_{w\in [\mathrm{conv}(F)]_\vee} (w_3+w_4-w_1-w_2)\geqslant 2$.

  • $\begingroup$ Your example appears to work, but I think an additional calculation is needed. $F$ only contains the 3 vectors $f,g,h$, yes? Your calculation shows supremacy for a given choice of $\sigma$, not for the average over $\sigma$. I performed the calculation, though, and there's a gap, so the example seems valid. $\endgroup$ – Aryeh Kontorovich May 1 at 14:40
  • $\begingroup$ The values that I get for the two Rademacher averages are 100 and 102, respectively -- not very large. $\endgroup$ – Aryeh Kontorovich May 1 at 14:49
  • $\begingroup$ But for any $\sigma$ we have an inequality (may be non strict) in the same direction. Thus equality could only appear if all of them turn into equalities, right? $\endgroup$ – Fedor Petrov May 1 at 15:22
  • $\begingroup$ Yes, point taken. So the question is, how much of a (multiplicative) gap can you force for a single $\sigma$? For what fraction of the $\sigma$'s? I've expanded the conjecture in case this challenge appeals to you :) $\endgroup$ – Aryeh Kontorovich May 1 at 15:47

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