Jan 4 (Reuters) - Australian shares are set to open lower on Tuesday, a day after many states in the country reported record numbers of COVID-19 daily cases and as the number of people hospitalised rose, hurting investor sentiment at the start of the new year. The local share price index futures dropped 1.2%, a 95.6-point discount to the underlying S&P/ASX 200 index (.AXJO) close. The benchmark slipped 0.9% on Dec. 31. The New Zealand stock market is closed on Tuesday for a public holiday.