In the warm humid room described, moisture from the air will be condensing on the ice. At 80% relative humidity, the dew point is about 27º C. This means than water will condense on any surface cooler than this. Ice will have a maximum surface temperature of about 0º C and will certainly be condensing moisture from the air.
Near the ice, the air will be less humid. Absolute humidity is a measure of the volume of water vapor in a given volume of air. In terms of absolute humidity there is less water in the air near the ice. We cannot use the term relative humidity here because relative humidity is related to how much water a volume of air can hold at a given temperature. The temperature is also going to be cooler near the ice. If one were to measure both the temperature and the humidity near the ice, the relative humidity reading might actually exceed 100%; meaning that the air is super-saturated.
Curious footnote: If the surface is very very cold (about the temperature of liquid nitrogen 77º K) water doesn't condense as quickly as the nitrogen and oxygen in the air. This is because the latent heat of vaporization of oxygen and nitrogen is about one tenth that of water vapor. You get some condensation of water (which turns quickly into ice). But near the surface the air might actually have a higher absolute humidity because the nitrogen and oxygen are being condensed quickly leaving the water vapor behind.