Kantar Worldpanel grocery-market data showing Tesco's best sales performance since June helped fragile sentiment in the battered supermarket industry to improve slightly on Tuesday.Shares of the three listed UK supermarket chains, Tesco, J Sainsbury and Wm Morrison, were all on the rise in afternoon trade, with the sector among the best performers in London.Kantar figures showed that the grocery market expanded by 0.4% in the 12 weeks to 12 October, compared with a year ago, even though like-for-like prices declined by 0.2%.This was the first time the market was in deflation since Kantar's recorded data began in October 2006 as the top-tier grocers continued to cut prices amid heightened competition from the so-called 'discounters'.Tesco, the biggest UK retailer, saw its market share decline to 28.8% from 30.1% a year earlier and "is yet to see substantial improvement", according to Kantar's consumer insight consultant Fraser McKevitt."However it seems it may be turning a corner as sales are down 3.6%, which is the grocer's best figure posted since June," he said, adding that trends at the grocer were "in the right direction".Tesco shares were up 3.1% at 184.78p by 14:24. The data came as welcome relief to investors of the retail giant, whose share price has been hammered over recent months by a string of profit warnings, accounting irregularities and several senior-management changes.Shares in Sainsbury and Morrisons were up 3% and 1.8% respectively on Tuesday.High-growth rivals Aldi and Lidl continued to build on their shares of the market, rising to 4.8% and 3.5% respectively.Even so, McKevitt said: "Watching the numbers very closely, you could say that Aldi and Lidl have peaked - it's been the fifth month in a row their combined growth rate has declined."Nevertheless, growing sales by a combined 23% year-on-year, the companies are still clearly outperforming the market and "that won't provide much comfort to their competitors", he said.Top performing sectors so far todayAutomobiles & Parts 7,264.37 +2.88%General Retailers 2,615.90 +2.32%Industrial Engineering 9,222.26 +2.27%Oil Equipment, Services & Distribution 21,011.91 +1.96%Food & Drug Retailers 2,605.88 +1.94%Bottom performing sectors so far todayTechnology Hardware & Equipment 997.48 -4.36%Household Goods & Home Construction 11,676.38 -0.62%Tobacco 38,663.38 -0.61%Pharmaceuticals & Biotechnology 11,746.22 -0.02%