Do large- and mid-caps try to shun low share prices? As NASDAQ and NYSE stocks must be priced at least $1 per share, it feels intuitive and doubtless that large- and mid-caps fancy their share prices to outstrip and remain well over \$1. But what minimum (range of) share price do they desire? To wit, at what share prices would they  think about reverse stock splitting to boost their share price? 

Why Some Companies Don't Split Their Stock explains why companies eschew low share prices.


  Why Not Split?
  
  There is not a whole lot of evidence to suggest that stock splits matter. Finance professors have examined stock splits and see no actual impact on a company’s value or performance.
  
  Many companies prefer to avoid splitting because they believe a high stock price gives the company a level of prestige. A company trading at $1,000 per share, for example, will be perceived as more valuable even though the firm's market capitalization may be the same as a company whose shares trade at \$50.
  
  Smaller companies may also wish to avoid stock splits because of a danger of share values falling too low. There have been cases where companies have split shares only to see the stock market dive, pushing shares below \$10. Psychologically, this may turn off some shareholders, and in extreme cases, share prices may be too low for a company to be listed on an exchange. Companies will avoid splitting to protect themselves from this possibility.
I have no expertise in this area whatsoever, but I've taken a quick look and see a significant body of research on reverse stock splits. Although there are many reasons that reverse stock splits occur, overall findings on the hypothesis of an optimal share price are inconclusive. 

Here is a review of 26 articles published about reverse stock splits in international journals from 1970 to 2010. I'll quote from the abstract:


  Reasons attributed by firms for initiating a reverse split are ‘image
  improvement’, ‘artificial increase of price’, ‘reduction of 
  stockholder servicing expenses’, ‘improving marketability’ ‘increase
  the stock's investor base’ ‘trading range hypothesis’ ‘signaling
  hypothesis’’ liquidity hypothesis’, ‘regulatory norms’ ‘facilitates
  going private’ and ‘compulsions from creditors, in case of a
  bankruptcy’. In the case of hypothesis testing, it is noticed that,
  researchers are not able to reach in a consensus on the outcome. 


Section 3.5 of this review focuses on "a widely accepted belief that forward stock splits and reverse splits are intended to keep the stock price within some 'optimal range'." Radcliffe and Gillespie (1979) "commented that reverse split cannot achieve an optimum range." Studies that focus on the Hong Kong market in particular do no support an optimum trading range hypothesis, at least not in the case of reverse stock splits. However Jain et al. (2004) "documented a significant increase in institutional ownership for stocks with prices of $5 after the reverse split." Jog and Zhu (2004) "found that institutional share holding increases after reverse split which helped them to conclude that an optimum price can reached (at least to institutional investors) through reverse split."

So while there are numerous reasons that companies may benefit from reverse splits, there is not strong evidence for the existence of an ideal share price range. If there were such a price range though, the study by Jain et al. suggests that the minimum might be around $5.