Stock Dilution of new shareholder, public offering

Hi all,

I just have a question regarding the share dilution when a company has issued new shares to the public.
While I was looking through some of the old IPO's in a market somewhere( is not important), I've seen a company offering its 33% of shares to the public.

The thing I quite did not get was, the book value of shares of the owner was 5 million (made up numbers) made up from 5 mil shares with nominal value of 1 each. With the new offering the new shares were determined at 2.81 per share, so he basically was offering the 33% of the company for 2.81 each.

The thing is we obviously have to assume the valuation must be correct so that the real value of the company is reflected in 2.81 per share. However, if it is not the case, then just by looking at the book value composition after the public offering, would not that be a dilution for the new shareholders? In other words based on pure book value numbers the old owner is getting 1.60 for each new share while new owners are paying a lot more at loss. ((5,000 * 1 + 2,500 *2.81/ 7500))

What I really mean is would that be a wise investment in terms of investor? I mean book value may nowhere near reflect the fair value however valuations could sometimes miss it completely right? If the share price is even 2.80 after IPO, well arent the owners diluted right after the public offering?