(1) Carr and Wu ("A Tale of Two Indices," http://www.math.nyu.edu/research/carrp/papers/pdf/vixov_florida3.pdf)
note that the square root of the price of forward variance is an upper bound for the fair value of VIX futures. This is because the price of VIX futures is the expected value of the square root of future expected variance and the expected value of the square root of a variable is smaller than or equal to the square root of the expected value of the variable (equal in this case to the price of forward variance). Dupire (bdupire@bloomberg.net) derives the concavity adjustment which needs to be substracted from the price of forward variance to arrive at the fair value of VIX futures.
(2) Demeterfi, K., E. Derman, M. Kamal J. Zhou, 1999, "More than You Ever Wanted to Know about Volatility Swaps," March 1999, Goldman Sachs Quantitative Strategies Research Notes.
(3) A complementary explanation is that the skew results from excess demand for insurance from institutional investors. See Whaley and Bollen " Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?" Journal of Finance April 2004, Vol.59 No.2.
(4) Originally, VIX was calculated from Black-Scholes implied volatilities of at-the-money or closest-to-the-money OEX options.
(5) The replication of volatility depends on a similar mix of contracts, but the option positions are not static.
(6) This result is reported by Jim Gatheral in "Fitting the Volatility Skew," Case Studies in Financial Modelling Course Notes, Lecture 2, Courant Institute of Mathematical Sciences, Fall 2002.
(7) Carr, Peter and R. Lee, January 2004, "At-the-money Implied as a Robust Approximation of the Volatility Swap Rate." Working Paper
(8) The current futures price F0 is the expected value of future futures prices, and payment of futures contracts is deferred. This implies that the expected values of FT - F0 and of 
Fi are 0 .
(9) Recent empirical evidence indicates that the risk premium for volatility is negative. Investors are willing to pay to hold volatility. The negative risk premium partly explains the persistent historical bias of VIX over realized volatility.
* The methodology of the CBOE Volatility Index is owned by CBOE and may be covered by one or more patents or pending patent applications.